Question

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{x}{y} - \frac{y}{3} \right) \frac{x^2}{16} + \frac{xy}{12} + \frac{y^2}{9}\]

Answer 1

In the given problem, we have to find the value of equation using identity

Given \[\left( \frac{x}{y} - \frac{y}{3} \right) \frac{x^2}{16} + \frac{xy}{12} + \frac{y^2}{9}\]

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

We can rearrange the `(x/4 - y/3) (x^2/16 + (xy)/12 + y^2/9)`as

` =(x/4 - y/3) ((x/4)^2 + (y/3)^2 + (x/4)(y/3))`

` = (x/4)^3 - (y/3)^3`

\[= \left( \frac{x}{4} \right) \times \left( \frac{x}{4} \right) \times \left( \frac{x}{4} \right) - \left( \frac{y}{3} \right) \times \left( \frac{y}{3} \right) \times \left( \frac{y}{3} \right)\]

\[ = \frac{x^3}{64} - \frac{y^3}{27}\]

Now substituting the value x=3, in  `x^3/64 - y^3/27`we get,

`= x^3/64 - y^3/27`

`= (3)^3/64 - (-1)^3/27`

` = 27/64 + 1/27`

Taking Least common multiple, we get 

` =(27 xx 27)/(64 xx 27) + (1 xx 64)/(27 xx 64)`

`=729/1728 + 64 /1728`

` =(729 + 64)/1728`

` = 793/1728`

Hence the Product value of `(x/4 - y/3)(x^2/16 + (xy)/12 + y^2/9)`is ` = 793/1728`.