Evaluate:
[(35)−2]−2\Big[\Big(\dfrac{3}{5}\Big)^{-2}\Big]^{-2}[(53)−2]−2
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As we know, for any non-zero rational number a
a−n=1ana^{-n} = \dfrac{1}{a^n}a−n=an1 and an=1a−na^{n} = \dfrac{1}{a^{-n}}an=a−n1.
[(35)−2]−2=[(53)2]−2=[(5×53×3)]−2=(259)−2=(925)2=9×925×25=81625\Big[\Big(\dfrac{3}{5}\Big)^{-2}\Big]^{-2}\\[1em] = \Big[\Big(\dfrac{5}{3}\Big)^2\Big]^{-2}\\[1em] = \Big[\Big(\dfrac{5 \times 5}{3 \times 3}\Big)\Big]^{-2}\\[1em] = \Big(\dfrac{25}{9}\Big)^{-2}\\[1em] = \Big(\dfrac{9}{25}\Big)^2\\[1em] = \dfrac{9 \times 9}{25 \times 25}\\[1em] = \dfrac{81}{625}[(53)−2]−2=[(35)2]−2=[(3×35×5)]−2=(925)−2=(259)2=25×259×9=62581
Hence, [(35)−2]−2=81625\Big[\Big(\dfrac{3}{5}\Big)^{-2}\Big]^{-2} = \dfrac{81}{625}[(53)−2]−2=62581
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