Question

type the correct answer in each box. what values of a and b make this equation true? \\((4 + \sqrt{-49}) - 2(\sqrt{(-4)^2} + \sqrt{-324}) = a + bi\\) \\(a = \square\\) \\(b = \square\\)

Explanation:

Step1: Simplify square roots of negative numbers

Recall that \(\sqrt{-n}=\sqrt{n}\cdot i\) for \(n>0\), and \(\sqrt{(-m)^2}=\vert m\vert\).
First, simplify each square root:

• \(\sqrt{-49}=\sqrt{49}\cdot i = 7i\)
• \(\sqrt{(-4)^2}=\sqrt{16}=4\)
• \(\sqrt{-324}=\sqrt{324}\cdot i = 18i\)

Step2: Substitute and expand the expression

Substitute these into the original equation:
\((4 + 7i)-2(4 + 18i)\)
Expand the second term: \(4 + 7i-8 - 36i\)

Step3: Combine like terms

Combine real parts and imaginary parts separately:
Real parts: \(4-8=-4\)
Imaginary parts: \(7i-36i=-29i\)
So the expression simplifies to \(-4-29i\), which is in the form \(a + bi\) where \(a=-4\) and \(b=-29\).

Answer:

\(a = -4\), \(b = -29\)