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

Represent complex numbers and their operations on the complex plane.

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

Perform arithmetic operations with complex numbers.

Perform operations on vectors.

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

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

Reason quantitatively and use units to solve problems.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Perform operations on matrices and use matrices in applications.

Extend the properties of exponents to rational exponents.
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: 5