To find the absolute value (or modulus) of the complex number [tex]\( -4 - \sqrt{2}i \)[/tex], we can use the formula for the absolute value of a complex number [tex]\( a + bi \)[/tex], which is given by:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
Here, [tex]\( a \)[/tex] is the real part and [tex]\( b \)[/tex] is the imaginary part. For the complex number [tex]\( -4 - \sqrt{2}i \)[/tex]:
- The real part [tex]\( a = -4 \)[/tex]
- The imaginary part [tex]\( b = -\sqrt{2} \)[/tex]
Step-by-step solution:
1. Square the real part:
[tex]\[ (-4)^2 = 16 \][/tex]
2. Square the imaginary part:
[tex]\[ (-\sqrt{2})^2 = (\sqrt{2})^2 = 2 \][/tex]
3. Add the squares of the real and imaginary parts:
[tex]\[ 16 + 2 = 18 \][/tex]
4. Take the square root of the sum to find the absolute value:
[tex]\[ \sqrt{18} \][/tex]
The result is approximately:
[tex]\[ \sqrt{18} \approx 4.242640687119285 \][/tex]
Thus, the absolute value of the complex number [tex]\( -4 - \sqrt{2}i \)[/tex] is [tex]\( 4.242640687119285 \)[/tex].
By comparing the result with the provided options, the absolute value does not correspond to any of the options directly. Therefore, among the given choices, we should acknowledge the absolute value as [tex]\( 4.242640687119285 \)[/tex] even though it doesn't match any of the given options exactly.
