Question

If x = −2 and y = 1, by using an identity find the value of the following

\[\left( 5y + \frac{15}{y} \right) \left( 25 y^2 - 75 + \frac{225}{y^2} \right)\]

Answer 1

Given \[\left( 5y + \frac{15}{y} \right) \left( 25 y^2 - 75 + \frac{225}{y^2} \right)\]

We shall use the identity `a^3 + b^3 = (a+b)(a^2 - ab + b^2)`,

We can rearrange the  \[\left( 5y + \frac{15}{y} \right) \left( 25 y^2 - 75 + \frac{225}{y^2} \right)\]as

` = (5y + 15/y)[(5y)^2 + (15/y)^2 - (5y) (15/y)]`

` = (5y)^3 + (15/y)^3`

 ` = (5y) xx (5y) xx (5y) + (15/y) xx (15/y) xx (15/y)`

` = 125y^3 + 3375/y^3`

Now substituting the value  y = 1in  `125y^3 + 3375/y^3`

` = 125y^3 + 3375/y^3`

 `= 125(1)^3 + 3375/(1)^3` 

`= 125 + 3375` 

` = 3500`

Hence the Product value of  \[\left( 5y + \frac{15}{y} \right) \left( 25 y^2 - 75 + \frac{225}{y^2} \right)\]is  3500.