If $x = 3$ and $y = -1$, find the values of each of the following using in identity:$ \left(\frac{5}{x}+5 x\right)\left(\frac{25}{x^{2}}-25+25 x^{2}\right) $

Given: 

$x = 3$ and $y = -1$

To do: 

We have to find the value of \( \left(\frac{5}{x}+5 x\right)\left(\frac{25}{x^{2}}-25+25 x^{2}\right) \).

Solution: 

We know that,

$a^{3}+b^{3}=(a+b)(a^{2}-a b+b^{2})$

$a^{3}-b^{3}=(a-b)(a^{2}+a b+b^{2})$

Therefore,

$(\frac{5}{x}+5 x)(\frac{25}{x^{2}}-25+25 x^{2})=(\frac{5}{x}+5 x)[(\frac{5}{x})^{2}-\frac{5}{x} \times 5 x+(5 x)^{2}]$

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

$=\frac{125}{x^{3}}+125 x^{3}$

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

$=\frac{125}{9}+125 \times 27$

$=\frac{125}{27}+3375$

$=\frac{125+91125}{27}$

$=\frac{91250}{27}$

Hence, $(\frac{5}{x}+5 x)(\frac{25}{x^{2}}-25+25 x^{2})=\frac{91250}{27}$.

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Updated on: 10-Oct-2022

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