Simplify each of the following:$(x+\frac{2}{x})^{3}+(x-\frac{2}{x})^{3}$

Given:

$(x+\frac{2}{x})^{3}+(x-\frac{2}{x})^{3}$

To do:

We have to simplify the given expression.

Solution:

We know that,

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

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

Therefore,

$(x+\frac{2}{x})^{3}+(x-\frac{2}{x})^{3}=[x^{3}+\frac{8}{x^{3}}+3 \times x^{2} \times \frac{2}{x}+3 \times x \times \frac{4}{x^{2}}]+[x^{3}-\frac{8}{x^{3}}-3 \times x^{2} \times \frac{2}{x}+3 \times x \times \frac{4}{x^{2}}]$

$=x^{3}+\frac{8}{x^{3}}+6 x+\frac{12}{x}+x^{3}-\frac{8}{x^{3}}-6 x+\frac{12}{x}$

$=2 x^{3}+\frac{24}{x}$

Hence, $(x+\frac{2}{x})^{3}+(x-\frac{2}{x})^{3}=2 x^{3}+\frac{24}{x}$.

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Updated on: 10-Oct-2022

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