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Simplify the following:$\frac{3^{-\frac{6}{7}} \times 4^{-\frac{3}{7}} \times 9^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$.
Given :
The given expression is $\frac{3^{-\frac{6}{7}} \times 4^{-\frac{3}{7}} \times 9^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$.
To do:
We have to simplify the given expression.
Solution:
$\frac{3^{-\frac{6}{7}} \times 4^{-\frac{3}{7}} \times 9^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$
We know that,
$(a^m)^n = a^{m\times n}$
$a^m \times a^n = a^{m+n}$
$\frac{a^m}{a^n} = a^{m-n}$
$\frac{3^{-\frac{6}{7}} \times 4^{-\frac{3}{7}} \times 9^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}} = \frac{3^{-\frac{6}{7}} \times (2^2)^{-\frac{3}{7}} \times (3^2)^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$
$= \frac{3^{-\frac{6}{7}} \times 2^{2\times{-\frac{3}{7}}} \times 3^{2\times{\frac{3}{7}}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$
$= \frac{3^{-\frac{6}{7}} \times 2^{-\frac{6}{7}} \times 3^{\frac{6}{7}} \times 2^{\frac{6}{7}}}{4 + 1 + \frac{1}{4}}$
$ = \frac{3^{-\frac{6}{7}+ \frac{6}{7}} \times 2^{-\frac{6}{7}+ \frac{6}{7}}}{5+ \frac{1}{4}}$
$ = \frac{3^0 \times 2^0}{ \frac{5\times 4 + 1}{4}}$
$ = \frac{1\times 1}{ \frac{21}{4}}$
$= \frac{4}{21}$
Therefore, the value of $\frac{3^{-\frac{6}{7}} \times 4^{-\frac{3}{7}} \times 9^{\frac{3}{7}} \times 2^{\frac{6}{7}}}{2^{2}+2^{0}+2^{-2}}$ is $\frac{4}{21}$
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