Multiple Choice
Simplify the root.
9. Roots, Radicals, and Complex Numbers
Introduction to Radical Expressions
Multiple Choice
Simplify the root.
A
B
C
D
Imaginary
Verified step by step guidance1
Recognize that the expression is the cube root of the square root of -125, which can be written as \(\sqrt{-125}^{\,\frac{1}{3}}\) or equivalently \(\left(\sqrt{-125}\right)^{\frac{1}{3}}\).
Rewrite the square root as an exponent: \(\sqrt{-125} = (-125)^{\frac{1}{2}}\), so the entire expression becomes \(\left((-125)^{\frac{1}{2}}\right)^{\frac{1}{3}}\).
Use the power of a power property: \(\left(a^{m}\right)^{n} = a^{m \times n}\), so the expression simplifies to \((-125)^{\frac{1}{2} \times \frac{1}{3}} = (-125)^{\frac{1}{6}}\).
Express -125 as \(-1 \times 125\) to separate the negative sign: \((-1)^{\frac{1}{6}} \times 125^{\frac{1}{6}}\).
Since 125 is \$5^3$, rewrite \(125^{\frac{1}{6}}\) as \(\left(5^3\right)^{\frac{1}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}} = \sqrt{5}\). Consider the nature of \((-1)^{\frac{1}{6}}\) to determine if the result is real or imaginary.
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