Rewrite expressions with radicals and rational exponents using exponent properties.
Decide whether a radical expression and a rational-exponent expression are equivalent
Problem
Are \(\sqrt{x^{3}}\) and \(x^{3/2}\) equivalent for \(\text{all}~\text{real}~x\) in their shared domain?
Big Picture
What this problem is really about
Equivalence of radical and exponent forms must be judged on their shared real domain. We’ll determine where the whole radicand is nonnegative, rewrite the square root as a one-half power, multiply the nested exponents, and verify that the equality includes the domain endpoint.
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