If x2-1-x2-51- find the value of x3-1-x3-

X^2+1/x^2=51 (x 1/x )^2=x^2+1/x^2 2 =51 2=49=(7)^2 ∴ x 1/x=7 Cubing both sides (x 1/x )^3=(7)^3 ⇒ x^3 1/x^3 3(x 1/x )=343 ⇒ x^3 1/x^3 3× 7 = 343 ⇒ x^3 1/x^3 21 ...

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Byju's Answer

Standard IX

Mathematics

Algebraic Identities

If x2+1/x2=51...

Question

If $x^{2} + \frac{1}{x^{2}} = 51$ , find the value of $x^{3} - \frac{1}{x^{3}}$ .

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Solution

$x^{2} + \frac{1}{x^{2}} = 51$

${(x - \frac{1}{x})}^{2} = x^{2} + \frac{1}{x^{2}} - 2 = 51 - 2 = 49 = (7)^{2} ∴ x - \frac{1}{x} = 7 Cubing both sides {(x - \frac{1}{x})}^{3} = (7)^{3} \Rightarrow x^{3} - \frac{1}{x^{3}} - 3 (x - \frac{1}{x}) = 343 \Rightarrow x^{3} - \frac{1}{x^{3}} - 3 \times 7 = 343 \Rightarrow x^{3} - \frac{1}{x^{3}} - 21 = 343 \Rightarrow x^{3} - \frac{1}{x^{3}} = 343 + 21 = 364 ∴ x^{3} - \frac{1}{x^{3}} = 364$

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If x2+1x2=51, find the value of x3−1x3.

x2+1x2=51 (x−1x)2=x2+1x2−2=51−2=49=(7)2∴ x−1x=7Cubing both sides(x−1x)3=(7)3⇒ x3−1x3−3(x−1x)=343⇒ x3−1x3−3×7=343⇒ x3−1x3−21=343⇒ x3−1x3=343+21=364∴ x3−1x3=364

If $x^{2} + \frac{1}{x^{2}} = 51$ , find the value of $x^{3} - \frac{1}{x^{3}}$ .