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Factorize each of the following expressions:
\( \frac{x^{3}}{216}-8 y^{3} \)
Given:
\( \frac{x^{3}}{216}-8 y^{3} \)
To do:
We have to factorize the given expression.
Solution:
We know that,
$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$
$a^3 - b^3 = (a - b) (a^2 + ab + b^2)$
Therefore,
$\frac{x^{3}}{216}-8 y^{3}=(\frac{x}{6})^{3}-(2 y)^{3}$
$=(\frac{x}{6}-2 y)[(\frac{x}{6})^{2}+\frac{x}{6} \times 2 y+(2 y)^{2}]$
$=(\frac{x}{6}-2 y)(\frac{x^{2}}{36}+\frac{x y}{3}+4 y^{2})$
Hence, $\frac{x^{3}}{216}-8 y^{3}=(\frac{x}{6}-2 y)(\frac{x^{2}}{36}+\frac{x y}{3}+4 y^{2})$.
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