(i) (−24332)−35\Big(-\dfrac{243}{32}\Big)^{-\dfrac{3}{5}}(−32243)−53
(ii) (52364)−23\Big(5\dfrac{23}{64}\Big)^{-\dfrac{2}{3}}(56423)−32
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Given,
⇒(−24332)−35⇒(−3525)−35⇒[(−32)5]−35⇒(−32)−5×35⇒(−32)−3⇒(−23)3⇒−827.\Rightarrow \Big(-\dfrac{243}{32}\Big)^{-\dfrac{3}{5}}\\[1em] \Rightarrow \Big(-\dfrac{3^5}{2^5}\Big)^{-\dfrac{3}{5}}\\[1em] \Rightarrow \Big[\Big(-\dfrac{3}{2}\Big)^5\Big]^{-\dfrac{3}{5}}\\[1em] \Rightarrow \Big(-\dfrac{3}{2}\Big)^{-\dfrac{5 \times 3}{5}}\\[1em] \Rightarrow \Big(-\dfrac{3}{2}\Big)^{-3}\\[1em] \Rightarrow \Big(-\dfrac{2}{3}\Big)^3\\[1em] \Rightarrow -\dfrac{8}{27}.⇒(−32243)−53⇒(−2535)−53⇒[(−23)5]−53⇒(−23)−55×3⇒(−23)−3⇒(−32)3⇒−278.
Hence, (−24332)−35=−827\Big(-\dfrac{243}{32}\Big)^{-\dfrac{3}{5}} = -\dfrac{8}{27}(−32243)−53=−278.
(ii) Given,
⇒(52364)−23⇒(34364)−23⇒(7343)−23⇒[(74)3]−23⇒(74)−2×33⇒(74)−2⇒(47)2⇒1649.\Rightarrow \Big(5\dfrac{23}{64}\Big)^{-\dfrac{2}{3}}\\[1em] \Rightarrow \Big(\dfrac{343}{64}\Big)^{-\dfrac{2}{3}}\\[1em] \Rightarrow \Big(\dfrac{7^3}{4^3}\Big)^{-\dfrac{2}{3}}\\[1em] \Rightarrow \Big[\Big(\dfrac{7}{4}\Big)^3\Big]^{-\dfrac{2}{3}}\\[1em] \Rightarrow \Big(\dfrac{7}{4}\Big)^{-\dfrac{2 \times 3}{3}}\\[1em] \Rightarrow \Big(\dfrac{7}{4}\Big)^{-2}\\[1em] \Rightarrow \Big(\dfrac{4}{7}\Big)^2\\[1em] \Rightarrow \dfrac{16}{49}.⇒(56423)−32⇒(64343)−32⇒(4373)−32⇒[(47)3]−32⇒(47)−32×3⇒(47)−2⇒(74)2⇒4916.
Hence, (52364)−23=1649\Big(5\dfrac{23}{64}\Big)^{-\dfrac{2}{3}} = \dfrac{16}{49}(56423)−32=4916.
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