Question

using rational exponents to rewrite expressions - item 50622
which expression has the same meaning as ( p^{\frac{11}{5}} )?
options: ( sqrt11{p^5} ), ( \frac{p^{11}}{p^5} ), ( sqrt5{p^{11}} ), ( (p^{11})(p^{\frac{1}{5}}) )

Explanation:

Step1: Recall the formula for rational exponents

The formula for converting a rational exponent \(a^{\frac{m}{n}}\) to a radical is \(\sqrt[n]{a^m}\) (where \(n\) is the index of the root and \(m\) is the power of the base inside the radical).

Step2: Apply the formula to \(p^{\frac{11}{5}}\)

For the expression \(p^{\frac{11}{5}}\), using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\), here \(a = p\), \(m = 11\), and \(n = 5\). So \(p^{\frac{11}{5}}=\sqrt[5]{p^{11}}\).

Answer:

\(\boldsymbol{\sqrt[5]{p^{11}}}\) (the option with \(\sqrt[5]{p^{11}}\))