Question

Re-express 2 √-45 as a complex number in the form a+bi.

Answer 1

To re-express \(2 \sqrt{-45}\) as a complex number in the form \(a + bi\), we first simplify the expression.

1. Start with the square root of \(-45\):

\[
\sqrt{-45} = \sqrt{45} \cdot \sqrt{-1}
\]

Here, \(\sqrt{-1}\) is represented as \(i\) (the imaginary unit).

2. Next, calculate \(\sqrt{45}\):

\[
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
\]

3. Now substitute back into the expression:

\[
\sqrt{-45} = 3\sqrt{5} \cdot i
\]

4. Multiply by 2 to get \(2 \sqrt{-45}\):

\[
2 \sqrt{-45} = 2 \cdot 3\sqrt{5} \cdot i = 6\sqrt{5} \cdot i
\]

5. Finally, express it in the form \(a + bi\):

\[
2 \sqrt{-45} = 0 + 6\sqrt{5} i
\]

Thus, the complex number in the form \(a + bi\) is:

\[
\boxed{0 + 6\sqrt{5} i}
\]