Question

use the rules for radicals to perform the indicated operation. assume all variables represent positive real numbers. $sqrt5{p}cdotsqrt5{p^{4}}$ select the correct choice below and, if necessary, fill in the answer box within your choice. a. $sqrt5{p}cdotsqrt5{p^{4}}$ simplifies to (simplify your answer. type an exact answer, using radicals as needed.) b. the expression cannot be simplified.

Explanation:

Step1: Apply product - rule of radicals

The product - rule of radicals states that $\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{ab}$ for $n\geq2$ and $a,b\geq0$ when $n$ is even. Here, $n = 5$, $a = p$, and $b=p^{4}$. So, $\sqrt[5]{p}\cdot\sqrt[5]{p^{4}}=\sqrt[5]{p\times p^{4}}$.

Step2: Use exponent rule

According to the exponent rule $a^{m}\cdot a^{n}=a^{m + n}$, when $a = p$, $m = 1$, and $n = 4$, we have $p\times p^{4}=p^{1 + 4}=p^{5}$. Then $\sqrt[5]{p\times p^{4}}=\sqrt[5]{p^{5}}$.

Step3: Simplify the radical

Since $\sqrt[n]{a^{n}}=a$ for all real $a$ when $n$ is odd, and here $n = 5$ (odd), $\sqrt[5]{p^{5}}=p$.

Answer:

A. $\sqrt[5]{p}\cdot\sqrt[5]{p^{4}}$ simplifies to $p$