For the imaginary unit i, where i^2 = -1, and any integer n, what is the value of the expr… — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , and any integer $n$ , what is the value of the expression $\frac{i^{4n+3} (1 + i)^8}{(1 - i)^6}$ ?

-2

To find the value of the expression, we simplify each part. First,

i^{4n+3}

simplifies to

-i

because

i^{4n} = 1

and

i^3 = -i

. Second,

(1+i)^8

is simplified by first squaring the base to get

(1+i)^2 = 2i

, and then raising the result to the fourth power:

(2i)^4 = 16

. Third,

(1-i)^6

is simplified by first squaring the base to get

(1-i)^2 = -2i

, and then cubing the result:

(-2i)^3 = -8i^3 = 8i

. Substituting these back into the expression yields

\frac{-i \cdot 16}{8i} = -2

. Thus, the expression simplifies to

-2

Simplify the term

i^{4n+3}

using the properties of powers of

i

i^{4n+3} = -i

Since

i^4 = 1

, we can rewrite

i^{4n+3}

(i^4)^n \cdot i^3 = 1^n \cdot (-i) = -i

Simplify the numerator term

(1+i)^8

(1+i)^8 = 16

We can rewrite

(1+i)^8

((1+i)^2)^4

. Since

(1+i)^2 = 1 + 2i + i^2 = 2i

, we have

(2i)^4 = 2^4 \cdot i^4 = 16 \cdot 1 = 16

Simplify the denominator term

(1-i)^6

(1-i)^6 = 8i

We can rewrite

(1-i)^6

((1-i)^2)^3

. Since

(1-i)^2 = 1 - 2i + i^2 = -2i

, we have

(-2i)^3 = (-2)^3 \cdot i^3 = -8 \cdot (-i) = 8i

Substitute the simplified components back into the original expression and divide.

\frac{-i \cdot 16}{8i} = -2

Substituting the terms gives

\frac{-i \cdot 16}{8i}

. The common factor of

i

in the numerator and denominator cancels out, and dividing

-16

8

yields

-2

Simplifying complex expressions involving powers of the imaginary unit and powers of complex binomials.

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