To determine the absolute value (or magnitude) of a complex number given in the form \( a + bi \), where \( a \) and \( b \) are real numbers, we use the formula:
[tex]\[
\sqrt{a^2 + b^2}
\][/tex]
For the complex number \( -4 - \sqrt{2} i \):
1. Identify the real part \( a \) and the imaginary part \( b \):
- The real part \( a = -4 \)
- The imaginary part \( b = -\sqrt{2} \)
2. Square the real and imaginary parts:
- \( (-4)^2 = 16 \)
- \( (-\sqrt{2})^2 = 2 \)
3. Sum the squares of the real and imaginary parts:
- \( 16 + 2 = 18 \)
4. Take the square root of the result:
- \( \sqrt{18} \)
To further simplify:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}
\][/tex]
Thus, the absolute value of the complex number \( -4 - \sqrt{2} i \) is:
\[
3\sqrt{2}
\
