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,

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

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

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

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

$\Rightarrow x^{3}+\frac{1}{x^{3}}=110$

Hence, the value of $x^{3}+\frac{1}{x^{3}}$ is 110.