Question

simplify the expression to ( a + bi ) form:
( -sqrt{36} + sqrt{-128} - sqrt{36} + sqrt{-200} )

Explanation:

Step1: Simplify real and imaginary parts

First, handle the real - valued square roots and the imaginary square roots separately. Recall that \(\sqrt{-x}=i\sqrt{x}\) for \(x>0\), and \(\sqrt{n^2}=n\) for \(n\geq0\).

For the real parts: We have \(-\sqrt{36}-\sqrt{36}\). Since \(\sqrt{36} = 6\), then \(-\sqrt{36}-\sqrt{36}=- 6-6=-12\).

For the imaginary parts: We have \(\sqrt{-128}+\sqrt{-200}\). Using the rule \(\sqrt{-x}=i\sqrt{x}\), we get \(i\sqrt{128}+i\sqrt{200}\).

Simplify \(\sqrt{128}\) and \(\sqrt{200}\):
\(\sqrt{128}=\sqrt{64\times2}=\sqrt{64}\times\sqrt{2}=8\sqrt{2}\)
\(\sqrt{200}=\sqrt{100\times2}=\sqrt{100}\times\sqrt{2}=10\sqrt{2}\)

So, \(i\sqrt{128}+i\sqrt{200}=i\times8\sqrt{2}+i\times10\sqrt{2}=(8\sqrt{2} + 10\sqrt{2})i\)

Step2: Combine like - terms in imaginary parts

Combine the coefficients of \(i\) in the imaginary parts. \(8\sqrt{2}+10\sqrt{2}=(8 + 10)\sqrt{2}=18\sqrt{2}\)

Step3: Combine real and imaginary parts

Now, combine the real part and the imaginary part. The real part is \(-12\) and the imaginary part is \(18\sqrt{2}i\). So the expression in \(a + bi\) form is \(-12+18\sqrt{2}i\)

Answer:

\(-12 + 18\sqrt{2}i\)