Question

simplify the expression to ( a + bi ) form:
( -sqrt{4} + sqrt{-80} - sqrt{1} + sqrt{-20} )
answer attempt 1 out of 3

Explanation:

Step1: Simplify real - part radicals

Simplify \(-\sqrt{4}\) and \(-\sqrt{1}\). We know that \(\sqrt{4} = 2\) and \(\sqrt{1}=1\). So \(-\sqrt{4}= - 2\) and \(-\sqrt{1}=-1\).
The expression becomes \(-2+\sqrt{-80}-1+\sqrt{-20}\).

Step2: Simplify imaginary - part radicals

Recall that \(\sqrt{-x}=\sqrt{x}\cdot i\) for \(x>0\).
For \(\sqrt{-80}\), we have \(\sqrt{-80}=\sqrt{80}\cdot i=\sqrt{16\times5}\cdot i = 4\sqrt{5}i\).
For \(\sqrt{-20}\), we have \(\sqrt{-20}=\sqrt{20}\cdot i=\sqrt{4\times5}\cdot i = 2\sqrt{5}i\).

Step3: Combine real and imaginary parts

Combine the real parts: \(-2-1=-3\).
Combine the imaginary parts: \(4\sqrt{5}i + 2\sqrt{5}i=6\sqrt{5}i\).

Step4: Write in \(a + bi\) form

The expression in \(a + bi\) form is \(-3+6\sqrt{5}i\).

Answer:

\(-3 + 6\sqrt{5}i\)