Concept used:

If a is real number n is a positive integer and \(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}\ \text{is a real number then the rational exponent expression}\ \displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}\ \text{is equivalent to radical expression}\ \displaystyle{a}^{{\frac{m}{{n}}}}.\)

The above statement can be express as,

\(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{{m}\text{/}{n}}}\)

Calculation:

The given expression is \(\displaystyle\sqrt{{{x}^{3}}}.\)

The property of nth \(\displaystyle{\sqrt[]{}}\) is,

\(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{{m}\text{/}{n}}}\)

Substitute 2 for n, 3 for m and x for a in the above equation.

\(\displaystyle{\sqrt[{{2}}]{{{x}^{3}}}}={\left({x}^{{{1}\text{/}{2}}}\right)}^{3}\)

\(\displaystyle={x}^{{{3}\text{/}{2}}}\)

Hence, the solution of the expression \(\displaystyle\sqrt{{{x}^{3}}}{i}{s}{x}^{{{3}\text{/}{2}}}\)

If a is real number n is a positive integer and \(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}\ \text{is a real number then the rational exponent expression}\ \displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}\ \text{is equivalent to radical expression}\ \displaystyle{a}^{{\frac{m}{{n}}}}.\)

The above statement can be express as,

\(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{{m}\text{/}{n}}}\)

Calculation:

The given expression is \(\displaystyle\sqrt{{{x}^{3}}}.\)

The property of nth \(\displaystyle{\sqrt[]{}}\) is,

\(\displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{{m}\text{/}{n}}}\)

Substitute 2 for n, 3 for m and x for a in the above equation.

\(\displaystyle{\sqrt[{{2}}]{{{x}^{3}}}}={\left({x}^{{{1}\text{/}{2}}}\right)}^{3}\)

\(\displaystyle={x}^{{{3}\text{/}{2}}}\)

Hence, the solution of the expression \(\displaystyle\sqrt{{{x}^{3}}}{i}{s}{x}^{{{3}\text{/}{2}}}\)