 # Radicals and Exponents Evaluate each expression: a) frac{sqrt{132}}{sqrt{3}} b) sqrt{2}sqrt{32} c) sqrt{frac{1}{4}}sqrt{frac{1}{64}} Bevan Mcdonald 2021-01-08 Answered
Radicals and Exponents Evaluate each expression:
a) $\frac{\sqrt{132}}{\sqrt{3}}$
b) $\sqrt{3}\left\{2\right\}\sqrt{3}\left\{32\right\}$
c) $\sqrt{4}\left\{\frac{1}{4}\right\}\sqrt{4}\left\{\frac{1}{64}\right\}$
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a) Formula used:
Power of nth roots:
$\sqrt{n}\left\{\frac{a}{b}\right\}=\frac{\sqrt{n}\left\{a\right\}}{\sqrt{n}\left\{a\right\}}$
Where n is any positive integer and a and b are bases.
Calculation:
The given exponential expression is $\frac{\sqrt{132}}{\sqrt{3}}$
Use the above-mentioned formula and calculate the value of $\frac{\sqrt{132}}{\sqrt{3}}$ as shown below.
$\frac{\sqrt{132}}{\sqrt{3}}=\frac{\sqrt{11\cdot 2\cdot 2\cdot 3}}{\sqrt{3}}$
$=\frac{2\sqrt{11}\cdot \sqrt{3}}{\sqrt{3}}$
$=2\sqrt{11}$
Thus, the value of exponential is $2\sqrt{11}$
b) Calculation:
Use the above mentioned formula and simplify the given expression as shown below.
$\sqrt{3}\left\{2\right\}\sqrt{3}\left\{32\right\}=\sqrt{3}\left\{2\right\}\sqrt{3}\left\{2\cdot 2\cdot 2\cdot 2\cdot 2\right\}$
$={\left({2}^{6}\right)}^{\frac{1}{3}}$
$={2}^{2}$
$=4$
Thus, the value of exponential expression is 4.
c) Calculation:
$\sqrt{4}\left\{\frac{1}{4}\right\}\sqrt{4}\left\{\frac{1}{64}\right\}=\sqrt{4}\left\{\frac{1}{4}\right\}\sqrt{4}\left\{\frac{1}{4\cdot 4\cdot 4}\right\}$
$=\sqrt{4}\left\{\frac{1}{{\left(4\right)}^{4}}\right\}$
$=\frac{1}{{\left({4}^{4}\right)}^{\frac{1}{4}}}$
$=\frac{1}{4}$
The value of exponential is $\frac{1}{4}.$
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