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Chapter 2: Problem 41

Perform the indicated operations and write the result in standard form. $$\sqrt{-8}(\sqrt{-3}-\sqrt{5})$$

Short Answer

Expert verified
-2\sqrt{6} - 2\sqrt{10}i

Step by step solution

01

Rewrite the square roots of negative numbers in terms of \( i \)

To start, rewrite the square roots of negative numbers in terms of \( i \), where \( i \) represents the square root of -1. So, express \(\sqrt{-8}\) and \(\sqrt{-3}\) in terms of \( i \). \[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i \] \[ \sqrt{-3} = \sqrt{3} \cdot \sqrt{-1} = i\sqrt{3} \]
02

Substitute these expressions back into the original expression

Once these radical expressions are written in terms of \( i \), substitute these expressions back into the original expression to obtain: \[ 2\sqrt{2}i(i\sqrt{3} - \sqrt{5}) \]
03

Simplify the Expression

Distribute the \( 2\sqrt{2}i \) into the parentheses \[ =2\sqrt{2}i(i\sqrt{3}) - 2\sqrt{2}i(\sqrt{5}) = 2\sqrt{6}i^2 - 2\sqrt{10}i \] Remember that \( i^2 \) equals -1, so replace \( i^2 \) with -1 and simplify to obtain the final answer.
04

Final Simplification

Replace \( i^2 \) with -1 and multiply: \[ = -2\sqrt{6} - 2\sqrt{10}i \] This is the result in standard form.

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