If 3x - 4x\dfrac{4}{x}x4 = 4 and x ≠ 0; find : 27x3−64x327x^3 - \dfrac{64}{x^3}27x3−x364.
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Given,
⇒3x−4x=4\Rightarrow 3x - \dfrac{4}{x} = 4⇒3x−x4=4
Cubing both sides we get :
⇒(3x−4x)3=43⇒(3x)3−(4x)3−3×3x×4x×(3x−4x)=64⇒(3x)3−(4x)3−36×4=64⇒27x3−64x3−144=64⇒27x3−64x3=64+144⇒27x3−64x3=208.\Rightarrow \Big(3x - \dfrac{4}{x}\Big)^3 = 4^3 \\[1em] \Rightarrow (3x)^3 - \Big(\dfrac{4}{x}\Big)^3 - 3 \times 3x \times \dfrac{4}{x} \times \Big(3x - \dfrac{4}{x}\Big) = 64 \\[1em] \Rightarrow (3x)^3 - \Big(\dfrac{4}{x}\Big)^3 - 36 \times 4 = 64 \\[1em] \Rightarrow 27x^3 - \dfrac{64}{x^3} - 144 = 64 \\[1em] \Rightarrow 27x^3 - \dfrac{64}{x^3} = 64 + 144 \\[1em] \Rightarrow 27x^3 - \dfrac{64}{x^3} = 208.⇒(3x−x4)3=43⇒(3x)3−(x4)3−3×3x×x4×(3x−x4)=64⇒(3x)3−(x4)3−36×4=64⇒27x3−x364−144=64⇒27x3−x364=64+144⇒27x3−x364=208.
Hence, 27x3−64x3=208.27x^3 - \dfrac{64}{x^3} = 208.27x3−x364=208.
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