Let the complex number z be defined as z = (4 - 3i)(1 + 2i) + 5i^{14}, where i = \sqrt{-1}… — ACT

Let the complex number $z$ be defined as $z = (4 - 3i)(1 + 2i) + 5i^{14}$ , where $i = \sqrt{-1}$ . What is the real part of $z$ ?

Answer: 5

The real part of the complex number

z

5

First, expand the product

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

to get

4 + 8i - 3i - 6i^2

. Replacing

i^2

with

-1

gives

10 + 5i

. Next, simplify

5i^{14}

. Since

i^{14} = (i^4)^3 \cdot i^2 = 1^3 \cdot (-1) = -1

, the term becomes

-5

. Adding the components together gives

z = (10 + 5i) - 5 = 5 + 5i

. The real part of this complex number is

5

Expand the product of the complex binomials

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

10 + 5i

Applying the distributive property gives

4 + 8i - 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 + 5i + 6 = 10 + 5i

Simplify the power of the imaginary unit in

5i^{14}

-5

Since the powers of

i

cycle every 4 terms,

i^{14} = i^{12} \cdot i^2 = 1 \cdot (-1) = -1

. Therefore,

5i^{14} = 5(-1) = -5

Add the simplified terms together to find

z

5 + 5i

Adding the real and imaginary parts of the terms yields

z = (10 + 5i) + (-5) = 5 + 5i

Determine the real part of

z

A complex number is written in the form

a + bi

, where

a

represents the real part. For

5 + 5i

, the real part is

5

Complex multiplication and simplification of powers of the imaginary unit

Question