If x−2x=6x - \dfrac{2}{x} = 6x−x2=6, find the value of (x3−8x3)\Big(x^3 - \dfrac{8}{x^3}\Big)(x3−x38).
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Given,
⇒x−2x=6\Rightarrow x - \dfrac{2}{x} = 6⇒x−x2=6
Upon cubing both sides we get :
⇒(x−2x)3=63⇒(x)3−(2x)3−3×x×2x×(x−2x)=216⇒(x)3−(2x)3−6×6=216⇒x3−8x3−36=216⇒x3−8x3=216+36⇒x3−8x3=252.\Rightarrow \Big(x - \dfrac{2}{x}\Big)^3 = 6^3 \\[1em] \Rightarrow (x)^3 - \Big(\dfrac{2}{x}\Big)^3 - 3 \times x \times \dfrac{2}{x} \times \Big(x - \dfrac{2}{x}\Big) = 216 \\[1em] \Rightarrow (x)^3 - \Big(\dfrac{2}{x}\Big)^3 - 6 \times 6 = 216 \\[1em] \Rightarrow x^3 - \dfrac{8}{x^3} - 36 = 216 \\[1em] \Rightarrow x^3 - \dfrac{8}{x^3} = 216 + 36 \\[1em] \Rightarrow x^3 - \dfrac{8}{x^3} = 252.⇒(x−x2)3=63⇒(x)3−(x2)3−3×x×x2×(x−x2)=216⇒(x)3−(x2)3−6×6=216⇒x3−x38−36=216⇒x3−x38=216+36⇒x3−x38=252.
Hence, x3−8x3=252.x^3 - \dfrac{8}{x^3} = 252.x3−x38=252.
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If x−1x=5x - \dfrac{1}{x} = 5x−x1=5, find the value of (x3−1x3)\Big(x^3 - \dfrac{1}{x^3}\Big)(x3−x31).
If x+1x=4x + \dfrac{1}{x} = 4x+x1=4, find the values of:
(i) (x3+1x3)\Big(x^3 + \dfrac{1}{x^3}\Big)(x3+x31)
(ii) (x−1x)\Big(x - \dfrac{1}{x}\Big)(x−x1)
(iii) (x3−1x3)\Big(x^3 - \dfrac{1}{x^3}\Big)(x3−x31)
If a2+1a2=23a^2 + \dfrac{1}{a^2} = 23a2+a21=23, find the values of:
(i) (a+1a)\Big(a + \dfrac{1}{a}\Big)(a+a1)
(ii) (a3+1a3)\Big(a^3 + \dfrac{1}{a^3}\Big)(a3+a31)
