Simplify the root.
${}^3\sqrt{-125}$

• A. $5$
• B. $0.2$
• C. $-5$
• D. Imaginary

Answer: (C)

Verified step by step guidance

1. Recognize that the expression is the cube root of the square root of -125, which can be written as \(\sqrt[3]{\sqrt{-125}}\).
2. Rewrite the nested radicals as a single exponent: recall that \(\sqrt{x} = x^{\frac{1}{2}}\) and \(\sqrt[3]{x} = x^{\frac{1}{3}}\), so the expression becomes \((-125)^{\frac{1}{2} \times \frac{1}{3}} = (-125)^{\frac{1}{6}}\).
3. Since the base is negative, consider expressing -125 as \(-1 \times 125\) to separate the negative sign and the positive number: \((-1)^{\frac{1}{6}} \times 125^{\frac{1}{6}}\).
4. Evaluate \(125^{\frac{1}{6}}\) by recognizing that \$125 = 5^3$, so \(125^{\frac{1}{6}} = (5^3)^{\frac{1}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}} = \sqrt{5}\).
5. Address the negative base \((-1)^{\frac{1}{6}}\): since the exponent is a fraction with an even denominator, this root involves imaginary numbers, so the simplified form will include imaginary components.