Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. frac{sqrt[4]{x^{7}}}{sqrt[4]{x^{3}}}

Simplifying Expressions Involving Radicals Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers.
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$

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Step 1
For any rational exponent $$\displaystyle\frac{{m}}{{n}}$$ in lowest terms, where m and n are integers and $$\displaystyle{n}{>}{0},$$ we define
$$\displaystyle{a}^{{\frac{{m}}{{n}}}}={\left(\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}\right)}^{{{m}}}=\sqrt{{{n}}}{\left\lbrace{a}^{{{m}}}\right\rbrace}$$
If n is even, then we require that $$\displaystyle{a}\geq{0}$$
Step 2
Consider the given expression,
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$
By using the law of exponents,
$$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}={\frac{{{x}^{{{\frac{{{7}}}{{{4}}}}}}}}{{{x}^{{{\frac{{{3}}}{{{4}}}}}}}}}$$
Apply exponent rule: $$\displaystyle{\frac{{{x}^{{{a}}}}}{{{x}^{{{b}}}}}}={x}^{{{a}-{b}}}$$, we get
$$\displaystyle{\frac{{{x}^{{{\frac{{{7}}}{{{4}}}}}}}}{{{x}^{{{\frac{{{3}}}{{{4}}}}}}}}}={x}^{{{\frac{{{7}}}{{{4}}}}-{\frac{{{3}}}{{{4}}}}}}$$
$$\displaystyle={x}^{{{\frac{{{4}}}{{{4}}}}}}$$
$$\displaystyle={x}$$
Final Statement:
The simplified form of $$\displaystyle{\frac{{\sqrt{{{4}}}{\left\lbrace{x}^{{{7}}}\right\rbrace}}}{{\sqrt{{{4}}}{\left\lbrace{x}^{{{3}}}\right\rbrace}}}}$$ is $$\displaystyle{x}$$.