If $ x-\frac{1}{x}=5 $, find the value of $ x^{3}-\frac{1}{x^{3}} $.

Given:

\( x-\frac{1}{x}=5 \)

To do:

We have to find the value of \( x^{3}-\frac{1}{x^{3}} \).

Solution:

We know that,

$(a-b)^3=a^3 - b^3 - 3ab(a-b)$

Therefore,

$x-\frac{1}{x}=5$

Cubing both sides, we get,

$(x-\frac{1}{x})^{3}=(5)^{3}$

$\Rightarrow x^{3}-\frac{1}{x^{3}}-3\times x \times \frac{1}{x}(x-\frac{1}{x})=125$

$\Rightarrow x^{3}-\frac{1}{x^{3}}-3 \times 5=125$

$\Rightarrow x^{3}-\frac{1}{x^{3}}=125+15$

$\Rightarrow x^{3}-\frac{1}{x^{3}}=140$

Hence, the value of $x^{3}-\frac{1}{x^{3}}$ is 140.   

Tutorialspoint

Updated on: 10-Oct-2022

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