standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
This activity requires students to make sense of negative powers of 10 as repeated multiplication by
\(\frac{1}{10}\)
in order to distinguish between equivalent exponential expressions. If students have time, instruct them to write the other expressions in each table as a power of 10 with a single exponent as well. Look for students who have productive debate regarding the interpretation of, for example,
\((10^2)^{\text-3}\)
versus
\((10^{\text-2})^3\)
and ask them to share their reasoning later (MP3).
Launch
Arrange students in groups of 2. Give students 15 minutes of partner work time followed by a whole-class discussion. Ask students to explain their reasoning to their partner as they work. If there is disagreement, tell students to work to reach an agreement.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time.
Supports accessibility for: Organization; Attention
Student Facing
Match each exponential expression with an equivalent multiplication expression:
\(\left(10^2\right)^3\)
\(\left(10^2\right)^{\text -3}\)
\(\left(10^{\text -2}\right)^3\)
\(\left(10^{\text -2}\right)^{\text-3}\)
|                              \(\frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)} \boldcdot \frac{1}{(10 \boldcdot 10)}\)                               |
|------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|          \(\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\)          |
| \(\frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} } \boldcdot \frac{1}{ \frac{1}{10} \boldcdot \frac{1}{10} }\) |
|                                                        \((10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10)\)                                                         |
Write
\((10^2)^{\text-3}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(\frac{10^2}{10^5}\)
\(\frac{10^2}{10^{\text -5}}\)
\(\frac{10^{\text -2}}{10^5}\)
\(\frac{10^{\text -2}}{10^{\text -5}}\)
| \(\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10} }\) |
|------------------------------------------------------------------------------------------------------------------------------------------------------------|
|                                     \(\frac{10 \boldcdot 10}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}\)                                     |
|                          \(\frac{ \frac{1}{10} \boldcdot \frac{1}{10} }{ 10 \boldcdot 10\boldcdot 10\boldcdot 10\boldcdot 10 }\)                           |
|           \(\frac{ 10 \boldcdot 10 }{ \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10}\boldcdot \frac{1}{10}}\)            |
Write
\(\frac{10^{\text -2}}{10^{\text -5}}\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Match each exponential expression with an equivalent multiplication expression:
\(10^4 \boldcdot 10^3\)
\(10^4 \boldcdot 10^{\text -3}\)
\(10^{\text -4} \boldcdot 10^3\)
\(10^{\text -4} \boldcdot 10^{\text -3}\)
|                                \((10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot ( \frac{1}{10} \boldcdot \frac{1}{10}\boldcdot \frac{1}{10})\)                                 |
|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| \(\left(\frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left( \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right)\) |
|                 \(\left(\frac{1}{10}\boldcdot \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10}\right) \boldcdot \left(10 \boldcdot 10 \boldcdot 10\right)\)                 |
|                                               \((10 \boldcdot 10 \boldcdot 10 \boldcdot 10) \boldcdot (10 \boldcdot 10 \boldcdot 10)\)                                                |
Write
\(10^{\text-4} \boldcdot 10^3\)
as a power of 10 with a single exponent. Be prepared to explain your reasoning.
Student Response
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Student Facing
Are you ready for more?
Priya, Jada, Han, and Diego stand in a circle and take turns playing a game.
Priya says, SAFE. Jada, standing to Priya's left, says, OUT and leaves the circle. Han is next: he says, SAFE. Then Diego says, OUT and leaves the circle. At this point, only Priya and Han are left. They continue to alternate. Priya says, SAFE. Han says, OUT and leaves the circle. Priya is the only person left, so she is the winner.
Priya says, “I knew I’d be the only one left, since I went first.”
Record this game on paper a few times with different numbers of players. Does the person who starts always win?
Try to find as many numbers as you can where the person who starts always wins. What patterns do you notice?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may struggle to distinguish between
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. Breaking down each expression into parts that emphasize repeated multiplication will help to illustrate the difference. For the first expression,
\(10^{\text-2} = \frac{1}{10} \boldcdot \frac{1}{10}\)
. The outer exponent of 3 means that
\(\frac{1}{10} \boldcdot \frac{1}{10}\)
is multiplied repeatedly 3 times. Similarly for the second expression,
\(10^2 = 10 \boldcdot 10\)
and the outer exponent of
\(\text- 3\)
means that the
reciprocal
of
\(10 \boldcdot 10\)
is multiplied repeatedly 3 times. In the end, both expressions are equal to
\(10^{\text-6}\)
.
Activity Synthesis
Select students who had disagreements during the activity to share what they disagreed about and how they came to an agreement. Consider asking:
“How is
\((10^{\text-2})^3\)
different from
\((10^2)^{\text-3}\)
? How are they the same?”
“Do you agree or disagree? Why?”
“Could you restate __'s reasoning in a different way?”
It is important for students to understand that the exponent rules work even with negative exponents. To see why, the whole class discussion must make a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and
\(\frac{1}{10}\)
. Contrast the expanded version of
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
. For
\((10^{\text-2})^3\)
, there are 3 factors that are
\(10^{\text-2}\)
, where
\(10^{\text-2}\)
is two factors that are
\(\frac{1}{10}\)
, so
\(\displaystyle \left(10^{\text-2}\right)^3 = \left( \frac{1}{10} \boldcdot \frac{1}{10} \right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right)\left(\frac{1}{10} \boldcdot \frac{1}{10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
For
\((10^2)^{\text-3}\)
, there are 3 factors that are
\(\frac{1}{10^2}\)
, so
\(\displaystyle (10^2)^{\text-3} = \left( \frac{1}{10 \boldcdot 10} \right)\left(\frac{1}{10 \boldcdot 10}\right)\left(\frac{1}{10 \boldcdot 10}\right) = \frac{1}{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10} = 10^{\text-6}.\)
Both
\(\left(10^{\text-2}\right)^3\)
and
\(\left(10^2\right)^{\text-3}\)
are equal to
\(10^{\text-6}\)
in the same way that
\(\text-2 \boldcdot 3 = 2 \boldcdot \text-3 = \text-6\)
.
Listening, Conversing: MLR8 Discussion Supports.
As students share the disagreements they had with their partner, press for details in students’ statements by requesting that students elaborate on an idea or give an example of their process of reconciliation. For example, ask students, “How did you explain your reasoning to your partner in a way that convinced them?” In this discussion, it may be useful to revoice student ideas to model mathematical language use by restating a statement in order to clarify, apply appropriate language, or involve more students.
Design Principle(s): Maximize meta-awareness


aligns: no