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 \(\displaystyle{x}\).

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 \(\displaystyle{x}\).