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Mathematics

Simplify the following:

3−67×4−37×937×26722+20+2−2\dfrac{3^{-\dfrac{6}{7}} \times 4^{-\dfrac{3}{7}} \times 9^{\dfrac{3}{7}} \times 2^{\dfrac{6}{7}}}{2^2 + 2^0 + 2^{-2}}

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Answer

Given,

⇒3−67×4−37×937×26722+20+2−2⇒3−67×(22)−37×(32)37×2674+1+(12)2⇒3−67×(2)−67×(3)67×2674+1+(14)⇒3−67+67×(2)−67+67(16+4+14)⇒30×20214⇒1214⇒421.\Rightarrow \dfrac{3^{-\dfrac{6}{7}} \times 4^{-\dfrac{3}{7}} \times 9^{\dfrac{3}{7}} \times 2^{\dfrac{6}{7}}}{2^2 + 2^0 + 2^{-2}} \\[1em] \Rightarrow \dfrac{3^{-\dfrac{6}{7}} \times (2^2)^{-\dfrac{3}{7}} \times (3^2)^{\dfrac{3}{7}} \times 2^{\dfrac{6}{7}}}{4 + 1 + \Big(\dfrac{1}{2}\Big)^{2}} \\[1em] \Rightarrow \dfrac{3^{-\dfrac{6}{7}} \times (2)^{-\dfrac{6}{7}} \times (3)^{\dfrac{6}{7}} \times 2^{\dfrac{6}{7}}}{4 + 1 + \Big(\dfrac{1}{4}\Big)} \\[1em] \Rightarrow \dfrac{3^{-\dfrac{6}{7} + \dfrac{6}{7}} \times (2)^{-\dfrac{6}{7} + \dfrac{6}{7}}}{\Big(\dfrac{16 + 4 + 1}{4}\Big)} \\[1em] \Rightarrow \dfrac{3^0 \times 2^0}{\dfrac{21}{4}} \\[1em] \Rightarrow \dfrac{1}{\dfrac{21}{4}} \\[1em] \Rightarrow \dfrac{4}{21}.

Hence, 3−67×4−37×937×26722+20+2−2=421\dfrac{3^{-\dfrac{6}{7}} \times 4^{-\dfrac{3}{7}} \times 9^{\dfrac{3}{7}} \times 2^{\dfrac{6}{7}}}{2^2 + 2^0 + 2^{-2}} = \dfrac{4}{21}.

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