options: Compute fluently with multi-digit numbers and find common factors and multiples.

Perform operations on vectors.

Extend the properties of exponents to rational exponents.

Represent and model with vector quantities.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on matrices and use matrices in applications.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

Reason quantitatively and use units to solve problems.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of numbers to the system of rational numbers.
problem_activity: Task
For this task, the letter $i$ denotes the imaginary unit, that is, $i=\sqrt{-1}$.
For each integer $k$ from 0 to 8, write $i^k$ in the form $a+bi$.
Describe the pattern you observe, and algebraically prove your observation.  In particular, simplify $i^{195}$.
Write each of the following expression in the form $a+bi$:
$i^2 + i + 1$
$i^3 + i^2 + i +1$
$i^4 + i^3 + i^2 + i +1$
$i^5 + i^4 + i^3 + i^2 + i +1$
$i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
$i^7 + i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
$i^8 + i^7 + i^6 + i^5 + i^4 + i^3 + i^2 + i +1$
Describe the pattern you observe, and algebraically prove your observation.  In particular, compute
$$
i^{195}+i^{194}+\cdots+i^2+i+1.
$$


correct_option_index: 9