Multiple Choice
Determine if the given square root evaluates to a real number.
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 involves the fourth root of the square root of a negative number: \(^4\sqrt{\sqrt{-625}}\).
Rewrite the expression by combining the roots: since \(^4\sqrt{\sqrt{x}} = x^{\frac{1}{2} \times \frac{1}{4}} = x^{\frac{1}{8}}\), rewrite the expression as \((-625)^{\frac{1}{8}}\).
Note that \(-625\) is a negative number, and taking an even root (like the fourth root or eighth root) of a negative number is not defined in the real numbers; it involves imaginary or complex numbers.
Express \(-625\) as \(-1 \times 625\) to separate the negative sign: \((-1)^{\frac{1}{8}} \times 625^{\frac{1}{8}}\).
Since \((-1)^{\frac{1}{8}}\) represents an eighth root of \(-1\), which is a complex number, conclude that the simplified root is imaginary (not a real number).
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