For the imaginary unit i, where i^2 = -1, what is the real part of the complex number z = … — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , what is the real part of the complex number $z = (2 - i)^3 + 9i$ ?

Cevap: 2

The real part of the complex number is 2.

Expanding the expression

(2 - i)^3

yields

2 - 11i

. Adding

9i

gives

2 - 2i

. The real part of this complex number is the term without

i

, which is 2.

Expand the squared binomial

(2 - i)^2

3 - 4i

Apply the identity

(a - b)^2 = a^2 - 2ab + b^2

and substitute

i^2 = -1

Multiply the result of the square by

(2 - i)

to calculate

(2 - i)^3

2 - 11i

Distribute the terms

(3 - 4i)(2 - i) = 6 - 3i - 8i + 4i^2

and substitute

i^2 = -1

Add

9i

to the simplified cube to find the complex number

z

2 - 2i

Combine the imaginary components:

-11i + 9i = -2i

Identify the real part of

z

The real part of a complex number

a + bi

a

Expanding complex binomials and simplifying powers of the imaginary unit

Soru