Question

FACTORISE THE EXPRESSION
$\large\underline{\bf{ \frac{1}{8} {a}^{2} + \frac{1}{4} {a}^{2}b + \frac{1}{6} {ab}^{2} + \frac{1}{27} {b}^{3} }}$
​

Answer 1

$\mathbb\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}$

$\rule{300pt}{0.1pt}$

First method :-

$\begin{array}{l} \\ \\ \displaystyle \rm \frac{1}{8} a^{3}+\frac{1}{4} a^{2} b+\frac{1}{6} a b^{2}+\frac{1}{27} b^{3} \\ \\ \displaystyle \rm=\left(\frac{1}{2} a\right)^{3}+3\left(\frac{1}{2} a\right)^{2}\left(\frac{1}{3} b\right)+3\left(\frac{1}{2} a\right)\left(\frac{1}{3} b\right)^{2} +\left(\frac{1}{3} b\right)^{3} \\ \\ \displaystyle \rm=\left(\frac{1}{2} a+\frac{1}{3} b\right)^{3}\\ \\ \color{magenta}\boxed{ \displaystyle \rm=\left(\frac{1}{2} a+\frac{1}{3} b\right)\left(\frac{1}{2} a+\frac{1}{3}b\right)\left(\frac{1}{2} a+\frac{1}{3} b\right) }\end{array}$

Second method :-

We have,

$\[ \begin{array}{l} \displaystyle\rm \frac{1}{8} a^{3}+\frac{1}{4} a^{2} b+\frac{1}{6} a b^{2}+\frac{1}{27} b^{3} \\\\ \displaystyle\rm =\left(\frac{1}{2} a\right)^{3}+\left(\frac{1}{3} b\right)^{3}+\frac{1}{2} a b\left(\frac{1}{2} a+\frac{1}{3} b\right) \\\\ \displaystyle\rm =\left(\frac{1}{2} a\right)^{3}+\left(\frac{1}{3} b\right)^{3}+\frac{1}{2} \cdot a b\left(\frac{1}{2} a+\frac{1}{3} b\right) \\\\ \displaystyle\rm =\left(\frac{1}{2} a\right)^{3}+\left(\frac{1}{3} b\right)^{3}+3 \cdot\left(\frac{a}{2}\right)\left(\frac{b}{3}\right)\left(\frac{1}{2} a+\frac{1}{3} b\right) \\\\ \displaystyle\rm =\left(\frac{1}{2} a+\frac{1}{3} b\right)^{3}\\ \\ \color{orangered}\boxed{ \displaystyle\rm =\left(\frac{1}{2} a+\frac{1}{3} b\right)\left(\frac{1}{2} a+\frac{1}{3}b\right)\left(\frac{1}{2} a+\frac{1}{3} b\right) }\end{array} \]$

Third method :-

$\text{Let \( \rm\dfrac{1}{2} a=x \) and \( \rm \dfrac{1}{3} b= \) \( \rm y ; \),}$

then,

$\text{Given expression \( \rm=x^{3}+y^{3}+\dfrac{1}{4} \)}$

$\[ \begin{array}{l} \displaystyle\rm (2 x)^{2}(3 y)+\frac{1}{6}(2 x)(3 y)^{2} \\\\ \displaystyle\rm =x^{3}+y^{3}+3 x^{2} y+3 . x y^{2} \\ \\ \displaystyle\rm=(x+y)^{3} \end{array} \]$

[Putting the values of a, b, ]

$\begin{array}{l} \displaystyle\rm =\left(\frac{1}{2} a+\frac{1}{3} b\right)^{3} \\ \\ \color{violet}\boxed{ \displaystyle\rm =\left(\frac{1}{2} a+\frac{1}{3} b\right)\left(\frac{1}{2} a+\frac{1}{3}b\right)\left(\frac{1}{2} a+\frac{1}{3} b\right) } \end{array}$