Question

# Simplify expression and express your answer using rational indicators. Suppose all letters represent positive numbers: \frac{1}{\sqrt[5]{x^{3}}}ZSK

Simplify expression and express your answer using rational indicators. Suppose all letters represent positive numbers.
$$\frac{1}{\sqrt[5]{x^3}}$$

2021-08-03

Step 1
Concept used:
If a - a real number, n - positive integer
$$\sqrt[n]{a^m}={a}^{{{m}{n}}}$$
The above statement can be expressed as,
$$\displaystyle{\frac{{{1}}}{{{a}^{{{n}}}}}}={a}^{{-{n}}}$$
Step 2
The property of n the root after combine the $$\displaystyle{\frac{{{1}}}{{{a}^{{{n}}}}}}={a}^{{-{n}}}$$ and $$\sqrt[n]{a^m}={a}^{{\frac{{m}}{{n}}}}$$ is,
$$\displaystyle\frac{1}{\sqrt[n]{a^m}}={a}^{{-\frac{{m}}{{n}}}}$$
Substitute 5 for n, 3 for m and x for a in the above equation.
$$\frac{1}{\sqrt[5]{x^3}}={\frac{{{1}}}{{{\left({x}^{{\frac{{1}}{{5}}}}\right)}^{{{3}}}}}}$$
$$\displaystyle={\left({x}^{{\frac{{1}}{{5}}}}\right)}^{{-{3}}}$$
$$\displaystyle={x}^{{-\frac{{3}}{{5}}}}$$
Thus, the solution of the expression $$\frac{1}{\sqrt[5]{x^3}}$$ is $$\displaystyle{x}^{{-\frac{{3}}{{5}}}}$$