options: Apply and extend previous understandings of numbers to the system of rational numbers.

Perform arithmetic operations with complex numbers.

Use properties of rational and irrational numbers.

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

Use complex numbers in polynomial identities and equations.

Perform operations on matrices and use matrices in applications.

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

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

Represent and model with vector quantities.

Extend the properties of exponents to rational exponents.

Perform operations on vectors.

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

Represent complex numbers and their operations on the complex plane.

Reason quantitatively and use units to solve problems.
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: 1