For the imaginary unit i, where i^2 = -1, the complex number z is defined as z = \frac{(1 … — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = \frac{(1 + 2i)^3}{2 - i}$ . What is the absolute value of $z$ ?

Answer: 5

Answer

The complex number

z

simplifies to

-4 - 3i

. The absolute value of a complex number

a + bi

is defined as

\sqrt{a^2 + b^2}

, which gives

\sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = 5

. Alternatively, using properties of the modulus,

|z| = \frac{|1 + 2i|^3}{|2 - i|} = \frac{(\sqrt{1^2 + 2^2})^3}{\sqrt{2^2 + (-1)^2}} = \frac{(\sqrt{5})^3}{\sqrt{5}} = 5

Step-by-Step Solution

Expand the numerator

(1 + 2i)^3

-11 - 2i

To begin simplifying the complex fraction by expanding the cubed binomial in the numerator.

Divide the result by the denominator

2 - i

by multiplying both numerator and denominator by the complex conjugate

2 + i

-4 - 3i

To express the quotient in standard complex form

a + bi

Find the absolute value of the simplified complex number

z = -4 - 3i

To calculate the final magnitude of the complex number using

|a + bi| = \sqrt{a^2 + b^2}

Key Concept

Complex number operations, complex division, and absolute value (modulus) of complex numbers

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic