# To calculate the simplified value of the radical expression: \sqrt{20x^{3}}+\sqrt{45x^{3}}

To calculate: The simplified value of the radical expression $$\displaystyle\sqrt{{{20}{x}^{{{3}}}}}+\sqrt{{{45}{x}^{{{3}}}}}$$

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Step 1
Formula used:
$$\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$$
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 $$\displaystyle{5}{x}\sqrt{{{5}{x}}}$$

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