Find the value of $27x^3 + 8y^3$ if $3x + 2y = 14$ and $xy = 8$.

Given:

$3x + 2y = 14$ and $xy = 8$

To do:

We have to find the value of $27x^3 + 8y^3$.

Solution:

$3 x+2 y=14$

Cubing both sides, we get,

$(3 x+2 y)^{3}=(14)^{3}$

$(3 x)^{3}+(2 y)^{3}+3 \times 3 x \times 2 y(3 x+2 y)=2744$

$27 x^{3}+8 y^{3}+18 x y(3 x+2 y)=2744$

$27 x^{3}+8 y^{3}+18 \times 8 \times 14=2744$

$27 x^{3}+8 y^{3}+2016=2744$

$27 x^{3}+8 y^{3}=2744-2016$

$27 x^{3}+8 y^{3}=728$

The value of $27 x^{3}+8 y^{3}$ is $728$.