Simplify the following:
[8−43÷2−2]12\Big[8^{-\dfrac{4}{3}} ÷ 2^{-2}\Big]^{\dfrac{1}{2}}[8−34÷2−2]21
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Given,
⇒[8−43÷2−2]12=[(18)43÷(12)2]12=[(123)43÷(12)2]12=[123×43÷(122)]12=[124×22]12=(122)12=12.\Rightarrow \Big[8^{-\dfrac{4}{3}} ÷ 2^{-2}\Big]^{\dfrac{1}{2}} = \Big[\Big(\dfrac{1}{8}\Big)^{\frac{4}{3}} ÷ \Big(\dfrac{1}{2}\Big)^2 \Big]^{\dfrac{1}{2}} \\[1em] = \Big[\Big(\dfrac{1}{2^3}\Big)^{\dfrac{4}{3}} ÷ \Big(\dfrac{1}{2}\Big)^2 \Big]^{\dfrac{1}{2}} \\[1em] = \Big[\dfrac{1}{2^{3 \times \dfrac{4}{3}}} ÷ \Big(\dfrac{1}{2^2}\Big)\Big]^{\dfrac{1}{2}} \\[1em] = \Big[\dfrac{1}{2^4} \times 2^2\Big]^{\dfrac{1}{2}} \\[1em] = \Big(\dfrac{1}{2^2}\Big)^{\dfrac{1}{2}} \\[1em] = \dfrac{1}{2}.⇒[8−34÷2−2]21=[(81)34÷(21)2]21=[(231)34÷(21)2]21=[23×341÷(221)]21=[241×22]21=(221)21=21.
Hence, [8−43÷2−2]12=12\Big[8^{-\dfrac{4}{3}} ÷ 2^{-2}\Big]^{\dfrac{1}{2}} = \dfrac{1}{2}[8−34÷2−2]21=21.
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