To find the absolute value (or magnitude) of the given complex number [tex]\(-3 - 6i\)[/tex], we use the formula for the magnitude of a complex number [tex]\( z = a + bi \)[/tex], which is given by:
[tex]\[ |z| = \sqrt{a^2 + b^2} \][/tex]
Here, [tex]\(a = -3\)[/tex] and [tex]\(b = -6\)[/tex]. Plugging these values into the formula, we get:
[tex]\[ | -3 - 6i | = \sqrt{(-3)^2 + (-6)^2} \][/tex]
Squaring the real and imaginary parts:
[tex]\[ (-3)^2 = 9 \][/tex]
[tex]\[ (-6)^2 = 36 \][/tex]
Now, add these results together:
[tex]\[ 9 + 36 = 45 \][/tex]
Next, take the square root of the sum:
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \][/tex]
Using the approximation that [tex]\(\sqrt{5} \approx 2.236\)[/tex], we find:
[tex]\[ 3 \times 2.236 \approx 6.708 \][/tex]
The absolute value of [tex]\(-3 - 6i\)[/tex] is approximately [tex]\(6.708\)[/tex].
Now, let's compare this result with the given choices:
a. [tex]\(6.07\)[/tex]
b. [tex]\(6.17\)[/tex]
c. [tex]\(6.7\)[/tex]
d. [tex]\(6.71\)[/tex]
The closest value to [tex]\(6.708\)[/tex] among the choices is clearly [tex]\(6.71\)[/tex].
Therefore, the best answer is:
[tex]\[ \boxed{\text{d}}\][/tex]
