Evaluate :

(i) $\dfrac{0.8 \times 0.8 \times 0.8 + 0.5 \times 0.5 \times 0.5}{0.8 \times 0.8 - 0.8 \times 0.5 + 0.5 \times 0.5}$

(ii) $\dfrac{1.2 \times 1.2 + 1.2 \times 0.3 + 0.3 \times 0.3}{1.2 \times 1.2 \times 1.2 - 0.3 \times 0.3 \times 0.3}$

(i) On taking,

a = 0.8 and b = 0.5

The expression $\dfrac{0.8 \times 0.8 \times 0.8 + 0.5 \times 0.5 \times 0.5}{0.8 \times 0.8 - 0.8 \times 0.5 + 0.5 \times 0.5}$ becomes :

$\Rightarrow \dfrac{a^3 + b^3}{a^2 - ab + b^2} \\[1em] \Rightarrow \dfrac{(a + b)(a^2 -ab + b^2)}{(a^2 - ab + b^2)} \\[1em] \Rightarrow a + b \\[1em] \Rightarrow 0.8 + 0.5 \\[1em] \Rightarrow 1.3$

Hence, $\dfrac{0.8 \times 0.8 \times 0.8 + 0.5 \times 0.5 \times 0.5}{0.8 \times 0.8 - 0.8 \times 0.5 + 0.5 \times 0.5}$ = 1.3

(ii) On taking,

a = 1.2 and b = 0.3

The expression $\dfrac{1.2 \times 1.2 + 1.2 \times 0.3 + 0.3 \times 0.3}{1.2 \times 1.2 \times 1.2 - 0.3 \times 0.3 \times 0.3}$ becomes :

$\Rightarrow \dfrac{a^2 + ab + b^2}{a^3 - b^3} \\[1em] \Rightarrow \dfrac{a^2 + ab + b^2}{(a - b)(a^2 + ab + b^2)} \\[1em] \Rightarrow \dfrac{1}{a - b} \\[1em] \Rightarrow \dfrac{1}{1.2 - 0.3} \\[1em] \Rightarrow \dfrac{1}{0.9} \\[1em] \Rightarrow \dfrac{10}{9} \\[1em] \Rightarrow 1\dfrac{1}{9}.$

Hence, $\dfrac{1.2 \times 1.2 + 1.2 \times 0.3 + 0.3 \times 0.3}{1.2 \times 1.2 \times 1.2 - 0.3 \times 0.3 \times 0.3} = 1\dfrac{1}{9}$.