Evaluate:
[(−34)−2]2\Big[\Big(-\dfrac{3}{4}\Big)^{-2}\Big]^2[(−43)−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.
[(−34)−2]2=[(−43)2]2=[(−4×(−4)3×3)]2=[(169)]2=(16×169×9)=(25681)=31381\Big[\Big(-\dfrac{3}{4}\Big)^{-2}\Big]^2\\[1em] = \Big[\Big(\dfrac{-4}{3}\Big)^2\Big]^2\\[1em] = \Big[\Big(\dfrac{-4 \times (-4)}{3 \times 3}\Big)\Big]^2\\[1em] = \Big[\Big(\dfrac{16}{9}\Big)\Big]^2\\[1em] = \Big(\dfrac{16 \times 16}{9 \times 9}\Big)\\[1em] = \Big(\dfrac{256}{81}\Big)\\[1em] = 3\dfrac{13}{81}[(−43)−2]2=[(3−4)2]2=[(3×3−4×(−4))]2=[(916)]2=(9×916×16)=(81256)=38113
Hence, [(−34)−2]2=31381\Big[\Big(-\dfrac{3}{4}\Big)^{-2}\Big]^2 = 3\dfrac{13}{81}[(−43)−2]2=38113
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