To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that make the equation true, let's solve the components step by step.
1. Calculate [tex]\( \sqrt{-49} \)[/tex]:
[tex]\[
\sqrt{-49} = 7i
\][/tex]
2. Calculate [tex]\( \sqrt{(-4)^2} \)[/tex]:
[tex]\[
\sqrt{(-4)^2} = \sqrt{16} = 4
\][/tex]
3. Calculate [tex]\( \sqrt{-324} \)[/tex]:
[tex]\[
\sqrt{-324} = 18i
\][/tex]
Now, substitute these values back into the expression:
[tex]\[
(4 + \sqrt{-49}) - 2 \left( \sqrt{(-4)^2} + \sqrt{-324} \right)
\][/tex]
This simplifies to:
[tex]\[
(4 + 7i) - 2 \left( 4 + 18i \right)
\][/tex]
Next, evaluate the components inside the parentheses:
[tex]\[
2 \left( 4 + 18i \right) = 2 \cdot 4 + 2 \cdot 18i = 8 + 36i
\][/tex]
Now, perform the subtraction:
[tex]\[
(4 + 7i) - (8 + 36i)
\][/tex]
Separate the real and imaginary parts:
[tex]\[
4 - 8 + 7i - 36i = -4 - 29i
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
\begin{array}{l}
a = -4 \\
b = -29
\end{array}
\][/tex]
