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a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
how do i calculate the probablity of getting a royal flush in the game of poker ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
how many triangles can be drawn by joining the vertices and centre of regular hexagon ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough .
why is the part of the question : `` sorted however the layer chooses '' significant ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
there 's only one 9 of spades.. so how can we choose 9 of spade first in the first selection and then in the secend selection 9 of spade last ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find ...
if we pick a spade , the probability of picking another spade increases , why do n't we have to worry about that affecting our result ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
what is an undefined value ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't ma...
surely if the order does n't matter then one can include all numbers with and without order ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
it should be 9 cards numbered from 1-9 right ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
then , after the 1st card is chosen , for the second card are n't we have 36-4=32 choices left ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards .
i had a question on my statistics homework that neither myself or my teacher could figure out the correct answer to : how many combinations are possible when choosing a hand of five cards from a standard deck of cards consisting of four cards from one suit and one card from another suit ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so at first we 'll say , well look , i have nine slots in my hand , right ? 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . right ?
what is the probability of getting a multiple of 4 ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
so what is the probability of a diamond flush ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
and then i have another . so then i have cards one , two , three , four , five , six , seven , eight . i have eight other cards .
what fraction of turns played on the machine will show five similar symbols in a row ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
we have 36 things . we chose 9 . so this numerator over here , this was 36 factorial .
why do i have to divide by 9 ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so let 's think about it . there are 36 unique cards -- and i wo n't worry about , you know , there 's nine numbers in each suit , and there are four suits , 4 times 9 is 36 . but let 's just think of the cards as being 1 through 36 , and we 're going to pick nine of them .
by the way the question is asked would n't 4 be an acceptable answer meaning you can make 4 nine-card hands from 36 cards ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
we chose 9 . so this numerator over here , this was 36 factorial . but 36 factorial would go all the way down to 27 , 26 , 25 . it would just keep going . but we stopped only nine away from 36 .
what would be the factorial of a negative number ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands . now , this would be true if order mattered . this would be true if i have card 15 here .
in terms of combinations vs permutations how do i decide whether a question takes into account order ( is it usually stated in the question ) ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
what are the possible permutations of a , a , b , b in four slots ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so we 're overcounting . we 're counting all of the different ways that the same number of cards can be arranged . so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged .
the number of ways of dividing 2p items into two equal groups of p each , groups having distinct identity ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
how many different hands can one draw in a game of 7-card rummy ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
how many unique combinations are possible for a word with 2 different letters repeating twice in the same word ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
same question..if 3 cards were identical remaining 33 were unique..how does that change the answer ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't ma...
so you use permutations when order matters and combinations when order does n't matter ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
just curious , does anyone know what the card game in the problem is ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so this is 36 factorial , so this part right here , that part right there , is not just 36 factorial . it 's 36 factorial divided by 36 , minus 9 factorial . what is 36 minus 9 ? it 's 27 .
without simplifying 36*35*34*33 ... ... ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
every , i guess we could say , natural number less than 9 . this is called 9 factorial , and you express it as an exclamation mark . so if we want to think about all of the different ways that we can have all of the different combinations for hands , this is the number of hands if we cared about the order , but then we...
what exactly is the factorial exclamation point thingy ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that means i have 9 ways to put something in the first slot .
are there actually 94143280 different ways to split the 9 cards ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen .
what 's the difference between permutation and combination ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
if we were thinking of these as two different hands , because we have the exact same cards , but they 're in different order , then what i just calculated would make a lot of sense , because we did it based on order . but they 're telling us that the cards can be sorted however the player chooses , so order does n't ma...
i mean does the order in which the cards are grouped matter ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
and then for the third slot , 34 , and then it just keeps going . then 33 to pick from , 32 , 31 , 30 , 29 , and 28 . so you might want to say that there are 36 times 35 , times 34 , times 33 , times 32 , times 31 , times 30 , times 29 , times 28 possible hands .
what is the probability that the sum is 30 ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
and this right here is called 36 choose 9 . and sometimes you 'll see this formula written like this , n choose k. and they 'll write the formula as equal to n factorial over n minus k factorial , and also in the denominator , k factorial . and this is a general formula that if you have n things , and you want to find ...
is it possible to have n choose k if k is larger than n ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
so in order to not overcount , we have to divide this by the ways in which nine cards can be rearranged . so we have to divide this by the way nine cards can be rearranged . so how many ways can nine cards be rearranged ? if i have nine cards and i 'm going to pick one of nine to be in the first slot , well , that mean...
does n't it matter which way the cards were dealt , and how many people were playing ?
a card game using 36 unique cards , four suits , diamonds , hearts , clubs and spades -- this should be spades , not spaces -- with cards numbered from 1 to 9 in each suit . a hand is chosen . a hand is a collection of 9 cards , which can be sorted however the player chooses . fair enough . how many 9 card hands are po...
fair enough . how many 9 card hands are possible ? so let 's think about it .
how many permutations can you make that are odd numbers ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
what do you do when you have a subtraction sign and addition sign ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
why does parentheses also work as a multiplication sign ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
when using the distributive property on a problem like 2 ( 4+b ) , would your answer be 8+2b or would it be 6b ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
is it possible to have the distributive law of division over addition/subtraction/multiplication ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
why when using the distributive law we have to multiply vs any of the other concepts ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
so we have 45 minus -- what 's 5 times 4 ? well , that 's 20 . 45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this ...
would n't ( 45-20 ) be the real answer insted of 25 ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 .
like is n't distrubitve property meaning only mutiple the numbers in the parenthiess only ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
so it 's 5 times 5 . 5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property .
what can be the way of finding the value of the variable is unknown ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
what is the difference between pemdas and bodmas ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . you could just evaluate 9 minus 4 and then multiply that times 5 .
what happens if i do n't know a number ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 . notice , we distributed the 5 . we multiplied it times both the 9 and the 4 .
what would happen if the distributed number was a letter ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
what was the simplified expression ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
is ( a multiplication sign ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
what is the point of using distributive property if you can just use pemdas ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
45 minus 20 , and that is equal to 25 , so this is using the distributive property right here . if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 .
how would you use two digit numbers ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
why did sal use the dot does that mean mulitpycation ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
how do you use the distributive property when dealing with fractions ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
why is the mutiplucation symble a dot not a `` x '' ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
if we did n't want to use the distributive property , if we just wanted to evaluate what 's in the parentheses first , we would have gotten -- let 's go in this direction -- 5 times -- what 's 9 minus 4 ? 9 minus 4 is 5 . let me do that in a different color .
when sal does 5 ( 9-4 ) why is n't a multiplication sighn next to the perenthasies ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
5 times 5 is just 25 , so we get the same answer either way . this is using the distributive law of multiplication over subtraction , usually just referred to as the distributive property . this is evaluating the inside of the parentheses first and then multiplying by 5 .
what is the reason of the distributive law ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
it what scenario would i ever want to use the distributive law of multication ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
how many multiplication signs are there ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it .
what does the question mean by simplify ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify .
what if 3 variables are unknown ?
rewrite the expression five times 9 minus 4 -- that 's in parentheses -- using the distributive law of multiplication over subtraction . then simplify . so let me just rewrite it . this is going to be 5 times 9 minus 4 , just like that . now , if we want to use the distributive property , well , you do n't have to . yo...
you could just evaluate 9 minus 4 and then multiply that times 5 . but if you want to use the distributive property , you distribute the 5 . you multiply the 5 times the 9 and the 4 , so you end up with 5 times 9 minus 5 times 4 .
also , what does `` distribute '' literally mean ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
well , the answer to that is not really known . one thing that is known is you 're more likely to have a phobia if you have a family member with a phobia . another way someone might develop a phobia , though , is from specific bad experiences .
can a word be a phobia ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
what 's the fear of parasites called ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
is acrophobia ( fear of heights ) common ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
for example , someone that suffered a severe bite by a rabid dog might develop cynophobia or fear of dogs . or even seeing someone else being very afraid of something could trigger a specific phobia . those that have a phobia will likely be diagnosed by a healthcare professional that asks specific questions about your ...
what is the phobia of being afraid of singing ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
this might lead to physical symptoms like sweating , muscle control issues , or fast heart rates . even though certain things like spiders can cause anxiety with a lot of people , it 's super important to emphasize the difference between an everyday anxiety about something and a specific phobia . as an example , an eve...
anxiety disorder is same as phobia ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
this avoidance might interfere with your social life and your relationship with your friends . obviously , a fear of moths is n't the only phobia , though . it 's not even in the top 10 .
how do people get a fear of moths ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
it 's not even in the top 10 . anyway , some of the more common phobias are things like fear of blood , or hemophobia , as well as fears of medical procedures , especially things like injections and needles . there can also be fears of animals , though , especially snakes or ophidiophobia , dogs or cynophobia , and spi...
what is the name for the fear of sharp things ?
phobia is this extreme and unreasonable or irrational fear of something . that something can be anything from an animal to an inanimate object , to situations or activities . having a phobia is n't just your everyday worries , stress , or fears though . it 's not just being scared of a dark alley or being worried abou...
oftentimes they 're fearful of not being able to escape . someone might also have a fear of heights or falling which is acrophobia , or a fear of flying , aerophobia . and finally , you could be afraid of some natural phenomenon , too , like lightning , which is called astraphobia .
if someone had a family member with a fear , would another family member develop a fear of the same thing , or is it possible to develop a fear of something else because a family member had a fear ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
dr. harris : it makes christ 's coming necessary to redeem us , but it also makes necessary the church that st. peter found . sometimes mary and christ are seen as the second adam and eve . adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something th...
why does eve make an effort to 'cover ' herself , but adam only covers his face ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
there is a physicality here , there 's a sense of weight and there 's a sense of musculature that i ca n't remember seeing in earlier painting . dr. harris : masaccio 's employing modeling very clearly from light to dark . he 's so interested in modeling because that 's what makes the forms appear three dimensional and...
what is `` modeling '' ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
adam and eve who caused the fall into sin and mary and christ who make possible salvation . dr. zucker : that idea is something that everybody in this church would be familiar with . i love the architecture on the extreme left , the gate of heaven itself , that they 've just left , reminds me of the indebtedness that m...
since many people were illiterate during this period , would these paintings have been considered a tool to help people grasp what they have leaned orally ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
one is that you can really see the giornata . you can see that adam was painted separately from eve and you can see the darker blue and back of adam that really highlight those different patches of plaster . dr. harris : those were not differentiatable in the 15th century .
why is there a different shade around adam and eve ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
dr. zucker : they are , they 're moving forward . dr. harris : masaccio is first artist in a very long time to attempt to paint the human body naturalistically . dr. zucker : yup .
did the artist intend that ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
dr. zucker : right , no that 's changed over time . dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work .
does oxidized silver remain silver or does oxidation convert it to a different element ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
dr. harris : by giornata you mean that the different days , the different parts of the fresco were painted in ? dr. zucker : right , giornata means a days work . dr. harris : this is buon fresco , which means that it was painted onto wet plaster and so an artist could only do a small section at a time because the plast...
why would n't masaccio do a better job of hiding the seams of the giornata around the contours of the figures instead of so obvious a halo surrounding them ?
( piano playing ) dr. zucker : in the brancacci chapel in santa maria del carmine just to the left of masaccio 's great painting the tribute of money is another painting by masaccio , the expulsion from eden . dr. harris : the fresco 's in this chapel all tell the story of the life of st. peter except for the expulsion...
his body is exposed to us and actually that 's interesting . this whole chapel was fairly recently cleaned and for a very long time there was a vine that covered up his genitals . dr. harris : that someone had painted over it .
why was the nudity covered up ?
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impression...
male : in fact , art historians think that perhaps the woman who 's seated is mary cassatt 's sister , lydia . female : degas and mary cassatt were friends for 40 years and degas actually was the one who suggested that mary cassatt exhibit with the impressionist group , which she did . she was the only american to do s...
why is it that monet , picasso , degas , and van gogh are artists that everybody knows something about but artists like mary cassat , winslow homer , casper david friedrich , and rosa bonheur are just as good but less well known ( i for one had n't heard of any of them til i watched these ) ?
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impression...
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day .
what causes an artist to become famous ?
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impression...
male : this is a painting that 's in some ways about the way we read the visual . it 's a painting about seeing . the subject is somebody looking at art as we , the viewers in the museum , are looking at art , so there 's this perfect replication .
what is the name of that painting ?
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day . we go to cafes . we go to performances . we sit in bars , we have a drink . we stroll down the boulevards . this is the stuff of modern life and the stuff of impression...
( jazzy music ) male : impressionism is so often painting of modern life , of activities that we engage in now . female : things that we do every day .
i may be jumping the gun here , but has the term `` impressionism '' come about by the time this painting was painted ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him . so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties .
why did meade remain so passive at such a crucial moment ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
do we have a `` fog of war '' moment with inadequate intelligence ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
and it 's extremely short , it 's only 272 words . now lincoln delivered the gettysburg address , on november 19th , 1863 . so it 's about three and a half months after the battle of gettysburg .
kim describes the individuals in the photograph taken at gettysburg on november 19 , 1863. who is the tall , bearded man in the top hat to the left of lincoln ( to the right of the viewer of the photograph ) ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
so the armies make kind of an incredible mess , and then they take off , leaving this tiny town of gettysburg , which has i think about 2,500 people , to deal with 50,000 casualties . so men who are dead , or wounded , maybe missing in action somewhere . and they really just do n't have the capacity for it , so the gov...
is there something that says somewhere that `` when you join the union , there is no way out , and if you try to we will fight you to the death until you give up and remain part of the union ... '' , perhaps in the constitution or otherwise ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
so gettysburg was this tremendously destructive battle with 50,000 casualties . and remember that , after the battle , lee is kinda fleeing for the life of his army . and not too long after that , meade pursues him .
but how did these leaders ( and the other important union and confederate army leaders ) relate in position and time to each other ?
alright , so we left off with the battle of gettysburg , from july 1st to 3rd , 1863 . and as i mentioned in the last video , gettysburg was a really significant battle in the civil war . it was a real turning point for the civil war , at which lee brought the forces of the south up into the north for a second attempt...
that 's cuz he worked really hard at it . so we 're fairly certain , that lincoln spent some time drafting the gettysburg address in the white house , long before he left . so , the day arrives , november 19th , 1863 , and everett gets set up in a tent , cuz he 's the real headliner of the day .
were there any slave-owners in the crowd at the time ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: what if we just went , you know , 4 steps ahead . how much rice would be here ? :4 steps ahead , so we 're going to multiple by 2 , then multiple by 2 again , then multiply by 2 again , the multiply by 2 again .
can anybody tell me how many actual grains of rice would be on the last ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: this is , this is interesting . this is also why when i look at a tree there 's thousands of leaves but just 1 trunk . and when you actually go up and you look inside the tree it only branches 3 or 4 times . : and that shows the power of exponential growth .
if a tree that splits twice is a binary tree , what is a tree that splits four times ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: well i guess why not ? it 's different . it 's not going to be 2 anymore .
are exponents the same as indices or is it completely different ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people .
why when i think of a mount everest sized mound of rice , it feels impossible ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that .
so the grains of rice and the game broad and the popsicle sticks all are divided by two ... 246810 ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: yeah , well , i mean , this one , the one you were making , right , every time you were kind of adding these popsicle sticks , you 're kind of branching out . 1 popsicle stick now becomes 2 popsicles sticks . then you keep doing that .
what is the point of the rice and popsicle sticks ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
and actually , it would be neat if there was a notation for that . : i did n't count this one out but it is the size of mount everest , the pile of rice . and it would feed 485 trillion people .
is n't mount everest too much for the pile of rice which is 2^63 ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks .
does any number to the 0 power equal 1 ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
1 popsicle stick becomes 2 but now you have 2 of them . so here you have 1 , now you have 1 times 2 . now each of these 2 branch into 2 , so now you have 2 times 2 , or you have 4 popsicle sticks .
i have to ask is it possible to divide a number by 1 ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree .
what is four squared to the 3rd power ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
so that is , take 63 , 2 's and multiply them all together . : this is the first square on our board . we have 1 grain of rice .
what would be the amount on the first square on the back of the board ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
9 , 2 's . 10 , 11 , 12 , 13 . so all of this stuff multiplied together .
why does a tree appear ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: hi sal . : hey britt .
what kind of drawing program does sal use ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: yeah , they have things to do . they have to go home and count grains of rice . : right .
how many grains of rice be on the chessboard if they finish putting all the rice ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: yeah , they have things to do . they have to go home and count grains of rice . : right .
how many grains of rice be on the chessboard if they finish putting all the rice ?
: hi sal . : hey britt . : how are you ? : good , looks like we have a game going on here . : not a game . yeah , kind of a challenge question for you . what i did is , i put 1 grain of rice in the first square . : that 's right . : there 's 64 squares on the board . : yup . : and in each consecutive square i ...
: and that 's what we have here . 1 times 2 times 2 times 2 , which is 8 . this is 2 to the third power . : when i see 2 to the power of something , let 's just say n. n could also be number of steps up this tree . i could think about it that way .
why could n't of they just stop after sal wrote 2 to the power of 63 ?