For the imaginary unit i, which of the following is equivalent to the complex expression \… — ACT

For the imaginary unit $i$ , which of the following is equivalent to the complex expression $\frac{2 + i^{15}}{1 - 2i^9}$ ?

\frac{4}{5} + \frac{3}{5}i

Simplifying the powers of

i

yields

i^{15} = -i

and

i^9 = i

, giving the expression

\frac{2 - i}{1 - 2i}

. Multiplying the numerator and denominator by the complex conjugate

1 + 2i

results in the fraction

\frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

. Expanding both parts and substituting

i^2 = -1

gives

\frac{4 + 3i}{5}

, which simplifies to

\frac{4}{5} + \frac{3}{5}i

Simplify the powers of the imaginary unit

i

in the expression.

i^{15} = i^{12} \cdot i^3 = 1 \cdot (-i) = -i

and

i^9 = i^8 \cdot i = 1 \cdot i = i

. The expression becomes

\frac{2 - i}{1 - 2i}

Reducing powers of

i

simplifies the expression and makes it easier to work with binomials.

Multiply the numerator and denominator by the complex conjugate of the denominator,

1 + 2i

\frac{2 - i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it to a real number.

Expand and simplify the numerator and denominator using the property

i^2 = -1

Numerator:

(2 - i)(1 + 2i) = 2 + 4i - i - 2i^2 = 2 + 3i - 2(-1) = 4 + 3i

. Denominator:

(1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 - 4(-1) = 5

Expanding the binomial products allows combining real and imaginary parts.

Write the resulting fraction in standard complex form

a + bi

\frac{4 + 3i}{5} = \frac{4}{5} + \frac{3}{5}i

Standard form separates the real part and the imaginary part clearly.

Simplifying complex expressions by reducing powers of

i

and rationalizing the denominator using the complex conjugate.

Question