To solve the given equation step by step, we need to simplify both terms involving complex numbers and real numbers.
Let's rewrite the equation and then simplify each part carefully:
[tex]\[
(4 + \sqrt{-49}) - 2\left(\sqrt{(-4)^2} + \sqrt{-324}\right) = a + bi
\][/tex]
### Step-by-Step Solution:
1. Simplify [tex]\(\sqrt{-49}\)[/tex]:
- [tex]\(\sqrt{-49}\)[/tex] can be written as [tex]\( \sqrt{49 \cdot -1} = \sqrt{49} \cdot \sqrt{-1} = 7i \)[/tex]
So, the first term becomes:
[tex]\[
4 + 7i
\][/tex]
2. Simplify [tex]\(\sqrt{(-4)^2}\)[/tex]:
- [tex]\(\sqrt{(-4)^2} = \sqrt{16} = 4\)[/tex]
3. Simplify [tex]\(\sqrt{-324}\)[/tex]:
- [tex]\(\sqrt{-324}\)[/tex] can be written as [tex]\( \sqrt{324 \cdot -1} = \sqrt{324} \cdot \sqrt{-1} = 18i \)[/tex]
4. Combine the simplified parts of the second term:
- [tex]\( \sqrt{(-4)^2} + \sqrt{-324} = 4 + 18i \)[/tex]
- Multiply by 2: [tex]\( 2 \times (4 + 18i) = 2 \times 4 + 2 \times 18i = 8 + 36i \)[/tex]
5. Subtract the second term from the first term:
- [tex]\((4 + 7i) - (8 + 36i)\)[/tex]
- Separate real and imaginary parts:
- Real part: [tex]\(4 - 8 = -4\)[/tex]
- Imaginary part: [tex]\(7i - 36i = -29i\)[/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -4
\][/tex]
[tex]\[
b = -29
\][/tex]
