Radicals and Exponents Evaluate each expression: a) frac{sqrt{132}}{sqrt{3}} b) sqrt[3]{2}sqrt[3]{32} c) sqrt[4]{frac{1}{4}}sqrt[4]{frac{1}{64}}

Question
Radicals and Exponents Evaluate each expression:
a) \(\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}\)
b) \(\displaystyle\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{32}\right\rbrace}\)
c) \(\displaystyle\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{64}}}}\right\rbrace}\)

Answers (1)

2021-01-09
a) Formula used:
Power of nth roots:
\(\displaystyle\sqrt{{{n}}}{\left\lbrace{\frac{{{a}}}{{{b}}}}\right\rbrace}={\frac{{\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}}}{{\sqrt{{{n}}}{\left\lbrace{a}\right\rbrace}}}}\)
Where n is any positive integer and a and b are bases.
Calculation:
The given exponential expression is \(\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}\)
Use the above-mentioned formula and calculate the value of \(\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}\) as shown below.
\(\displaystyle{\frac{{\sqrt{{{132}}}}}{{\sqrt{{{3}}}}}}={\frac{{\sqrt{{{11}\cdot{2}\cdot{2}\cdot{3}}}}}{{\sqrt{{{3}}}}}}\)
\(\displaystyle={\frac{{{2}\sqrt{{{11}}}\cdot\sqrt{{{3}}}}}{{\sqrt{{{3}}}}}}\)
\(\displaystyle={2}\sqrt{{{11}}}\)
Thus, the value of exponential is \(\displaystyle{2}\sqrt{{{11}}}\)
b) Calculation:
Use the above mentioned formula and simplify the given expression as shown below.
\(\displaystyle\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{32}\right\rbrace}=\sqrt{{{3}}}{\left\lbrace{2}\right\rbrace}\sqrt{{{3}}}{\left\lbrace{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\right\rbrace}\)
\(\displaystyle={\left({2}^{{{6}}}\right)}^{{{\frac{{{1}}}{{{3}}}}}}\)
\(\displaystyle={2}^{{{2}}}\)
\(\displaystyle={4}\)
Thus, the value of exponential expression is 4.
c) Calculation:
\(\displaystyle\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{64}}}}\right\rbrace}=\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}}}}\right\rbrace}\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{4}\cdot{4}\cdot{4}}}}\right\rbrace}\)
\(\displaystyle=\sqrt{{{4}}}{\left\lbrace{\frac{{{1}}}{{{\left({4}\right)}^{{{4}}}}}}\right\rbrace}\)
\(\displaystyle={\frac{{{1}}}{{{\left({4}^{{{4}}}\right)}^{{\frac{{{1}}}{{{4}}}}}}}}\)
\(\displaystyle={\frac{{{1}}}{{{4}}}}\)
The value of exponential is \(\displaystyle{\frac{{{1}}}{{{4}}}}.\)
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