Question

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

Rational exponents and radicals
ANSWERED
asked 2021-08-02

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

Expert Answers (1)

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 \(\displaystyle{x}^{{-\frac{{3}}{{5}}}}\)

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