Given that i = \sqrt{-1}, what is the value of the expression i^{83}? — ACT

Given that $i = \sqrt{-1}$ , what is the value of the expression $i^{83}$ ?

The value of the expression is

-i

The expression

i^{83}

can be simplified by dividing the exponent 83 by 4. Because the remainder is 3, the expression is equivalent to

i^3

. Since

i^3 = i^2 \cdot i = (-1) \cdot i = -i

, the correct value is

-i

Divide the exponent 83 by 4 to determine the remainder.

The quotient is 20 with a remainder of 3, which means

83 = 4 \times 20 + 3

Powers of the imaginary unit

i

repeat in a cycle of four:

i^1 = i

i^2 = -1

i^3 = -i

, and

i^4 = 1

Rewrite the expression using the rules of exponents and substitute the values of

i^4

and

i^3

i^{83} = i^{4(20) + 3} = (i^4)^{20} \cdot i^3 = (1)^{20} \cdot (-i) = -i

Since

i^4 = 1

, any power of

i

that is a multiple of 4 simplifies to 1, leaving only the remainder power to determine the final value.

Simplifying powers of the imaginary unit

i

Question