Step 1

Formula used:

The product rule for radicals,

\(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\)

Here, n is the positive integer and a and b are real number.

The relation between radical and rational exponent notation is expressed as,

\(\sqrt[n]{a}=a^{1/n}\)

Here, a is the radicand, and n is the index of the radical.

Step 2

Use the product rule for radicals to simplify the expression,

\(\displaystyle\sqrt{{{20}{x}^{{{3}}}}}+\sqrt{{{45}{x}^{{{3}}}}}=\sqrt{{{5}\cdot{4}\cdot\ {x}^{{{2}}}\cdot\ {x}}}+\sqrt{{{9}\cdot{5}\cdot\ {x}^{{{2}}}\cdot\ {x}}}\)

\(\displaystyle=\sqrt{{{5}{x}}}\cdot\sqrt{{{4}}}\cdot\sqrt{{{x}^{{{2}}}}}+\sqrt{{{9}}}\cdot\sqrt{{{5}{x}}}\cdot\sqrt{{{x}^{{{2}}}}}\)

\(\displaystyle=\sqrt{{{5}{x}}}\cdot\sqrt{{{2}^{{{2}}}}}\cdot\sqrt{{{x}^{{{2}}}}}+\sqrt{{{3}^{{{2}}}}}\cdot\sqrt{{{5}{x}}}\cdot\sqrt{{{x}^{{{2}}}}}\)

Use the relation between radical and rational exponent notation to simplify the expression,

\(\displaystyle\sqrt{{{20}{x}^{{{3}}}}}+\sqrt{{{45}{x}^{{{3}}}}}=\sqrt{{{5}{x}}}\cdot{\left({2}^{{{2}}}\right)}^{{\frac{{1}}{{2}}}}\cdot{\left({x}^{{{2}}}\right)}^{{\frac{{1}}{{2}}}}+{\left({3}^{{{2}}}\right)}^{{\frac{{1}}{{2}}}}\cdot\sqrt{{{5}{x}}}\cdot{\left({x}^{{{2}}}\right)}^{{\frac{{1}}{{2}}}}\)

\(\displaystyle=\sqrt{{{5}{x}}}\cdot{2}\cdot\ {x}+{3}\cdot\sqrt{{{5}{x}}}\cdot\ {x}\)

\(\displaystyle={5}{x}\sqrt{{{5}{x}}}\)

Hence, the simplified value of the provided expression is \(\displaystyle{5}{x}\sqrt{{{5}{x}}}\)