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}}}

Khaleesi Herbert 2020-12-30 Answered
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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Expert Answer

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Answered 2020-12-31 Author has 15482 answers
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 \(\displaystyle{x}\).
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