Multiple Choice
Simplify the root.
11. Roots, Radicals, and Complex Numbers
Introduction to Radical Expressions
Multiple Choice
Simplify the root.
A
B
C
D
Imaginary
Verified step by step guidance1
Identify the expression to simplify: the cube root of the square root of -125, which can be written as \(\sqrt[3]{\sqrt{-125}}\).
Rewrite the nested radicals using fractional exponents: the square root is the power of \(\frac{1}{2}\) and the cube root is the power of \(\frac{1}{3}\), so the expression becomes \(\left(-125\right)^{\frac{1}{2} \times \frac{1}{3}} = \left(-125\right)^{\frac{1}{6}}\).
Recognize that \(-125\) can be expressed as \(-1 \times 125\), and since \$125 = 5^3$, rewrite the base as \(-1 \times 5^3\).
Apply the exponent \(\frac{1}{6}\) to both parts: \(\left(-1\right)^{\frac{1}{6}} \times \left(5^3\right)^{\frac{1}{6}}\).
Simplify the powers: \(\left(5^3\right)^{\frac{1}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}}\), which is the square root of 5, and consider the nature of \(\left(-1\right)^{\frac{1}{6}}\) to determine if the result is real or imaginary.
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