# Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. NKS sqrt{x^{5}}x5

Question
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. NKS $$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}$$

2021-01-17
Step 1
"For any rational exponent $$\displaystyle\frac{{m}}{{n}}\ \frac{{m}}{{n}}$$ in lowest terms, where mm and nn are integers and $$\displaystyle{n}{>}{0},\ {n}{>}{0}$$ we define NKS $$\displaystyle{a}^{{\frac{{m}}{{n}}}}={\left(\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\right)}^{{{m}}}=\sqrt{{{n}}}{\left\lbrace{a}^{{{m}}}\right\rbrace}{a}\frac{{m}}{{n}}{\left({a}{n}\right)}{m}={a}{m}{n}$$
If nn is even, then we require that $$\displaystyle{a}\geq{0}{a}\geq{0}$$"
Step 2
$$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}$$
Consider the given expression,
$$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}={\left({x}^{{{5}}}\right)}^{{\frac{{1}}{{2}}}}{\left({x}{5}\right)}\frac{{1}}{{2}}$$
Apply Law of exponents $$\displaystyle{a}^{{\frac{{m}}{{n}}}}={\left(\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\right)}^{{{m}}}=\sqrt{{{n}}}{\left\lbrace{a}^{{{m}}}\right\rbrace}{a}\frac{{m}}{{n}}={\left({a}{n}\right)}{m}={a}{m}{n}$$ we get,
$$\displaystyle{\left({x}^{{{5}}}\right)}^{{\frac{{1}}{{2}}}}={x}^{{{\frac{{{5}}}{{{2}}}}}}{\left({x}{5}\right)}\frac{{1}}{{2}}={x}{52}$$
Therefore the expression $$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}$$ simplifies to $$\displaystyle{x}^{{{\frac{{{5}}}{{{2}}}}}}{x}{52}$$

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