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Chapter 2: Q34P (page 53)

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.
${(\frac{1 + i}{1 - i})}^{5}$

Short Answer

Expert verified

The answer is obtained ${|\frac{1 + i}{1 - i}|}^{5} = 1$

Step by step solution

01

Given information

It is given that $z = {|\frac{1 + i}{1 - i}|}^{5}$

02

Absolute value of the Complex number

Let z be a complex number consisting of two real numbers as:

$z = x + t y$

Here x and y are the real numbers.

The absolute value of z can be calculated as:

$| z | = \sqrt{x^{2} + y^{2}}$

03

Begin by stating the given information.

State the given information.

$z = {(\frac{1 + i}{1 - i})}^{5}$

04

Find the modulus of the function.

Find the modulus of the function.

$|z| = |{(\frac{1 + i}{1 - i})}^{5}| {|\frac{1 + i}{1 - i}|}^{5}$

The absolute value can be calculated as below:

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

Thus, the final answer is obtained as:

${|\frac{1 + i}{1 - i}|}^{5} = 1$

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Most popular questions from this chapter

Solve for all possible values of the real numbers $x$ and $y$ in the following equations. $(2 x - 3 y - 5) + i (x + 2 y + 1) = 0$

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