Simplify the root.
${}^4\sqrt{-625}$

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

Answer: (A)

Verified step by step guidance

1. Recognize that the expression is the fourth root of the square root of -625, which can be written as \(\sqrt[4]{\sqrt{-625}}\).
2. Rewrite the nested roots as a single root with a fractional exponent: \(\left(-625\right)^{\frac{1}{2} \times \frac{1}{4}} = \left(-625\right)^{\frac{1}{8}}\).
3. Since the base is negative, consider expressing -625 as \(-1 \times 625\) to separate the negative sign and the positive number.
4. Recall that \$625 = 5^4$, so rewrite the expression as \(\left(-1 \times 5^4\right)^{\frac{1}{8}}\).
5. Use the property of exponents to separate the terms: \(\left(-1\right)^{\frac{1}{8}} \times \left(5^4\right)^{\frac{1}{8}} = \left(-1\right)^{\frac{1}{8}} \times 5^{\frac{4}{8}} = \left(-1\right)^{\frac{1}{8}} \times 5^{\frac{1}{2}}\).