Question

Use the properties of complex numbers to simplify the following expression: (square root -25 + 2i) + 5i + ( 9 - square root -36)

Answer 1

To simplify the expression \((\sqrt{-25} + 2i) + 5i + (9 - \sqrt{-36})\), we will first express the square roots of the negative numbers using complex numbers.

1. Simplify \(\sqrt{-25}\):

\[
\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i
\]

2. Simplify \(\sqrt{-36}\):

\[
\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i
\]

Now, substituting these simplifications back into the original expression, we have:

\[
(5i + 2i) + 5i + (9 - 6i)
\]

Now, combine the terms:

1. Combine the \(i\) terms:

\[
5i + 2i + 5i - 6i = (5 + 2 + 5 - 6)i = 6i
\]

2. The real part is:

\[
9
\]

Now we can put both the real and the imaginary parts together:

\[
9 + 6i
\]

Thus, the simplified expression is:

\[
\boxed{9 + 6i}
\]