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]
