Simplify the following:
(27)2n3×(8)−n6(18)−n2\dfrac{(27)^{\dfrac{2n}{3}} \times (8)^{-\dfrac{n}{6}}}{(18)^{-\dfrac{n}{2}}}(18)−2n(27)32n×(8)−6n
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Given,
⇒(27)2n3×(8)−n6(18)−n2=(33)2n3×(23)−n6(2×32)−n2=32n×(2)−n2(2)−n2×(32)−n2=32n3−n=32n13n=32n×3n=3(2n+n)=33n.\Rightarrow \dfrac{(27)^{\dfrac{2n}{3}} \times (8)^{-\dfrac{n}{6}}}{(18)^{-\dfrac{n}{2}}} \\[1em] = \dfrac{(3^3)^{\dfrac{2n}{3}} \times (2^3)^{-\dfrac{n}{6}}}{(2 \times 3^2)^{-\dfrac{n}{2}}} \\[1em] = \dfrac{3^{2n} \times (2)^{-\dfrac{n}{2}}}{(2)^{-\dfrac{n}{2}} \times (3^2)^{-\dfrac{n}{2}}} \\[1em] = \dfrac{3^{2n}}{3^{-n}} = \dfrac{3^{2n}}{\dfrac{1}{3^n}} \\[1em] = 3^{2n} \times 3^n \\[1em] = 3^{(2n + n)} = 3^{3n}. \\[1em]⇒(18)−2n(27)32n×(8)−6n=(2×32)−2n(33)32n×(23)−6n=(2)−2n×(32)−2n32n×(2)−2n=3−n32n=3n132n=32n×3n=3(2n+n)=33n.
Hence, (27)2n3×(8)−n6(18)−n2=33n\dfrac{(27)^{\dfrac{2n}{3}} \times (8)^{-\dfrac{n}{6}}}{(18)^{-\dfrac{n}{2}}} = 3^{3n}(18)−2n(27)32n×(8)−6n=33n.
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