# Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. frac{sqrt[3]{8x^{2}}}{sqrt{x}}

Question
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$

2021-03-06
Step 1
$$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$
Law of exponents:
$$\displaystyle{a}^{{\frac{{m}}{{n}}}}={\left(\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\right)}^{{{m}}}=\sqrt{{{n}}}{\left\lbrace{a}^{{{m}}}\right\rbrace}$$
If n is even, then we require that $$\displaystyle{a}\geq{0}$$
Step 2
Consider the given expression,
1) $$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$
Apply radical rule: $$\displaystyle\sqrt{{{n}}}{\left\lbrace{a}{b}\right\rbrace}=\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\sqrt{{{n}}}{\left\lbrace{b}\right\rbrace}$$ in the numerator, we get
$$\displaystyle\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}=\sqrt{{{3}}}{\left\lbrace{8}\right\rbrace}\cdot\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}$$
$$\displaystyle=\sqrt{{{3}}}{\left\lbrace{2}^{{{3}}}\right\rbrace}\cdot\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}$$
$$\displaystyle={2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}$$
Substitute $$\displaystyle\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}={2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}$$ in equation (1), we get
$$\displaystyle{\frac{{{2}\sqrt{{{3}}}{\left\lbrace{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}={\frac{{{2}{x}^{{{\frac{{{2}}}{{{3}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{2}}}}}}}}}$$
Apply exponent rule: $$\displaystyle{\frac{{{x}^{{{a}}}}}{{{x}^{{{b}}}}}}={x}^{{{a}-{b}}}$$, we get
$$\displaystyle{\frac{{{2}{x}^{{{\frac{{{2}}}{{{3}}}}}}}}{{{x}^{{{\frac{{{1}}}{{{2}}}}}}}}}={2}{x}^{{{\frac{{{2}}}{{{3}}}}-{\frac{{{1}}}{{{2}}}}}}$$
$$\displaystyle={2}{x}^{{{\frac{{{1}}}{{{6}}}}}}$$
The simplified form of $$\displaystyle{\frac{{\sqrt{{{3}}}{\left\lbrace{8}{x}^{{{2}}}\right\rbrace}}}{{\sqrt{{{x}}}}}}$$ is $$\displaystyle{2}{x}^{{{\frac{{{1}}}{{{6}}}}}}$$

### Relevant Questions

Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$
Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. NKS $$\displaystyle\sqrt{{{x}^{{{5}}}}}{x}{5}$$
Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\frac{\sqrt[3]{8x^{2}}}{\sqrt{x}}$$
Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. $$\sqrt{x^{5}}$$
Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
a) $$r^{^1/_6}\ r^{^5/_6}$$
b) $$a^{^3/_5}\ a^{^3/_{10}}$$
Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle\sqrt{{{x}^{3}}}$$
Radical simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers.
a) $$\displaystyle\sqrt{{{6}}}{\left\lbrace{y}^{{{5}}}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{y}^{{{2}}}\right\rbrace}$$
b) $$\displaystyle{\left({5}\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}\right)}{\left({2}\sqrt{{{4}}}{\left\lbrace{x}\right\rbrace}\right)}$$
Combining radicals simplify the expression. Assume that all letters denote positive numbers.
$$\displaystyle\sqrt{{{16}{x}}}+\sqrt{{{x}^{5}}}$$
Simplify the expression, answering with rational exponents and not radicals. To enter $$\displaystyle{x}^{{\frac{{m}}{{n}}}},$$ type $$\displaystyle{x}^{{{\left(\frac{{m}}{{n}}\right)}}}.$$
$$\displaystyle\sqrt{{{3}}}{\left\lbrace{27}{m}^{{{2}}}\right\rbrace}=?$$
The given expression using rational exponents. Then simplify and convert back to radical notation. Assume that all variables represent positive real numbers.
Given:
The expression is $$\displaystyle\sqrt{{{81}{a}^{12}{b}^{20}}}$$
...