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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.
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