Simplify the following:
a) $ (0.3)^{3}+(0.2)^{3} $
b) $ \left(\frac{-3}{4}\right)^{3}-\left(\frac{-1}{4}\right)^{3} $
c) $ 1+\left(\frac{3}{5}\right)^{3} $

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,

a) $(0.3)^{3}+(0.2)^{3}=(0.3+0.2)[(0.3)^2-0.3\times0.2+(0.2)^2]$

$=(0.5)(0.09-0.06+0.04)$

$=0.05\times0.07$

$=0.0035$

b) $(\frac{-3}{4})^{3}-(\frac{-1}{4})^{3}=(\frac{-3}{4}+\frac{-1}{4})[(\frac{-3}{4})^2-\frac{-3}{4}\times\frac{-1}{4}+(\frac{-1}{4})^2]$

$=(\frac{-4}{4})(\frac{9}{16}-\frac{3}{16}+\frac{1}{16})$

$=(-1)(\frac{9-3+1}{16})$

$=(-1)\times\frac{7}{16}$

$=\frac{-7}{16}$

c) $1+(\frac{3}{5})^{3}=1^3+(\frac{3}{5})^{3}$

$=(1+\frac{3}{5})[(1)^2-1\times\frac{3}{5}+(\frac{3}{5})^2]$

$=(\frac{1\times5+3}{5})(1-\frac{3}{5}+\frac{9}{25})$

$=(\frac{5+3}{5})(\frac{1\times25-3\times5+9}{25})$

$=\frac{8}{5}\times(\frac{25-15+9}{25})$

$=\frac{8}{5}\times\frac{19}{25}$

$=\frac{154}{25}$

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Updated on: 10-Oct-2022

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