Question

simplify the radical expression.
\\(\sqrt5{-x^5}\\)
write your answer in the form a, \\(\sqrt5{b}\\), or \\(a\sqrt5{b}\\), where a and b are constants or expressions in x. use at most one radical in your answer, and at most one absolute value symbol in your expression for a.

Explanation:

Step1: Recall the property of nth roots

For an odd - numbered root (in this case, the 5th root, and 5 is odd), we have the property $\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{-a}=-\sqrt[n]{a}$ when $n$ is odd. Also, for any real number $a$, $\sqrt[n]{a^{n}} = a$ when $n$ is odd.

We are given the radical expression $\sqrt[5]{-x^{5}}$. We can rewrite $-x^{5}$ as $(-x)\times x^{4}$ or we can use the property of odd - numbered roots. Since $n = 5$ (odd), $\sqrt[5]{-x^{5}}=\sqrt[5]{(-x)^{5}}$.

Step2: Apply the property of odd - numbered roots

According to the property of odd - numbered roots, if $n$ is odd, then $\sqrt[n]{a^{n}}=a$. So, for $a=-x$ and $n = 5$, we have $\sqrt[5]{(-x)^{5}}=-x$.

Answer:

$-x$