For the imaginary unit i = \sqrt{-1}, the complex number w is defined as w = \frac{5 + 12i… — ACT

For the imaginary unit $i = \sqrt{-1}$ , the complex number $w$ is defined as $w = \frac{5 + 12i}{(1 - i)^4}$ . What is the absolute value of $w$ ?

Cevap: 3.25

The absolute value of

w

is 3.25.

The correct answer is 3.25 because simplifying the denominator yields

(1-i)^4 = -4

. Dividing the numerator by

-4

gives the complex number

w = -1.25 - 3i

. The absolute value of

w

is then calculated as

\sqrt{(-1.25)^2 + (-3)^2} = \sqrt{1.5625 + 9} = \sqrt{10.5625} = 3.25

. Alternatively, using properties of absolute values, the absolute value of the quotient is the quotient of the absolute values:

|w| = \frac{|5 + 12i|}{|1-i|^4} = \frac{\sqrt{5^2 + 12^2}}{(\sqrt{1^2 + (-1)^2})^4} = \frac{13}{4} = 3.25

Simplify the denominator

(1 - i)^4

(1 - i)^4 = -4

Calculate

(1 - i)^2 = -2i

, then square the result to obtain

(-2i)^2 = -4

Write the complex number

w

in standard form

a + bi

w = -1.25 - 3i

Divide each term in the numerator by the simplified denominator

-4

Calculate the magnitude

|w|

|w| = 3.25

Use the definition of absolute value of a complex number,

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

Absolute value of a complex number and operations on complex numbers

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