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Multiple Choice
Which expression is equivalent to sqrt{-80}?
A
8i sqrt{2}
B
sqrt{80}
C
-8 sqrt{10}
D
4i sqrt{5}
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Verified step by step guidance
1
Recognize that the expression \( \sqrt{-80} \) involves the square root of a negative number, which means it can be expressed using the imaginary unit \( i \), where \( i = \sqrt{-1} \).
Rewrite \( \sqrt{-80} \) as \( \sqrt{-1 \times 80} \), which can be separated into \( \sqrt{-1} \times \sqrt{80} \).
Replace \( \sqrt{-1} \) with \( i \), so the expression becomes \( i \times \sqrt{80} \).
Simplify \( \sqrt{80} \) by factoring 80 into its prime factors: \( 80 = 16 \times 5 \), so \( \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4 \sqrt{5} \).
Combine the terms to get the simplified expression: \( 4i \sqrt{5} \), which is equivalent to \( \sqrt{-80} \).