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What is the absolute value of the complex number [tex]-4-\sqrt{2} i[/tex]?

A. [tex]\sqrt{14}[/tex]

B. [tex]3 \sqrt{2}[/tex]

C. 14

D. 18

Sagot :

GinnyAnswer
To find the absolute value (or magnitude) of the complex number [tex]\( -4 - \sqrt{2}i \)[/tex], we use the formula for the magnitude of a complex number [tex]\( a + bi \)[/tex]. The formula is:

[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]

Given the complex number [tex]\( -4 - \sqrt{2}i \)[/tex], let's identify the real part [tex]\( a \)[/tex] and the imaginary part [tex]\( b \)[/tex]:

- The real part [tex]\( a \)[/tex] is [tex]\(-4\)[/tex]
- The imaginary part [tex]\( b \)[/tex] is [tex]\(-\sqrt{2}\)[/tex]

Next, we substitute these values into the formula:

[tex]\[ |-4 - \sqrt{2}i| = \sqrt{(-4)^2 + (-\sqrt{2})^2} \][/tex]

Calculate the squares of the real and imaginary parts:

[tex]\[ (-4)^2 = 16 \][/tex]

[tex]\[ (-\sqrt{2})^2 = 2 \][/tex]

Add these squared values together:

[tex]\[ 16 + 2 = 18 \][/tex]

Finally, take the square root of the sum:

[tex]\[ \sqrt{18} \][/tex]

We simplify [tex]\(\sqrt{18}\)[/tex] as follows:

[tex]\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \][/tex]

Thus, the absolute value of the complex number [tex]\( -4 - \sqrt{2}i \)[/tex] is:

[tex]\[ 3\sqrt{2} \][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{3\sqrt{2}} \][/tex]
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