Question

Find the value of 64x³ − 125z³, if 4x − 5z = 16 and xz = 12.

Answer 1

From given problem we have to find the value of  ${\displaystyle 64x^3-125z^3}$

Given  ${\displaystyle (4x-5z)=16,xz=12}$

On cubing both sides of ${\displaystyle (4x-5z)=16}$ we get

${\displaystyle (4x-5z^3)={\left(16\right)}^3}$

We shall use identity  ${\displaystyle {(a-b)}^3=a^3-b^3-3ab(a-b)}$

${\displaystyle 4x^3-125z^3-3\left(4x\right)\left(5z\right)(4x-5z)=16\times 16\times 16}$

             ${\displaystyle 64x^3-125z^3-60\left(xz\right)\left(16\right)=4096}$

              ${\displaystyle 64x^3-125z^3-60\left(12\right)\left(16\right)=4096}$

                        ${\displaystyle 64x^3-125z^3-11520=4096}$

                                         ${\displaystyle 64x^3-125z^3=4096+11520}$

                                          ${\displaystyle 64x^3-125z^3=15616}$

Hence the value of  ${\displaystyle 64x^3-125z^3}$  is . 15616.