Select the expression that is equivalent to the expression below.
$\sqrt[6]{x^5}$
The goal is to determine the equivalent expression for \(\sqrt[6]{x^5}\). In radical expressions, a radical of the form \(\sqrt[n]{x^m}\) can be converted into an equivalent expression using rational exponents. Specifically, the root index \(n\) becomes the denominator of the exponent, and the power of the radicand \(m\) becomes the numerator of the exponent. The general conversion formula is \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\).
We identify the components of the given expression \(\sqrt[6]{x^5}\):
Applying the conversion rule \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), we substitute \(n=6\) and \(m=5\) into the formula:
\[\sqrt[6]{x^5} = x^{\frac{5}{6}}\]
Comparing our derived result \(x^{\frac{5}{6}}\) with the provided options:
The expression \(x^{\frac{5}{6}}\) matches the fourth option. Therefore, the equivalent expression is \(x^{\frac{5}{6}}\).
Understanding how to convert radicals to fractional exponents is essential for simplifying algebraic expressions and performing calculus operations like derivatives and integrals. For example, if you are modeling the growth of a population or the cooling of an object where the rate depends on a fractional power, you would use this conversion to handle the variable more easily in your equations. This foundational skill allows for smoother manipulation of complex mathematical models used in physics and engineering.