Question

Prove that $\frac{0.85\times 0.85\times 0.85+0.15\times 0.15\times 0.15}{0.85\times 0.85-0.85\times 0.15+0.15\times 0.15}=1$.

Answer 1

$$\begin{gathered} \mathrm{LHS}:\frac{0.85\times 0.85\times 0.85+0.15\times 0.15\times 0.15}{0.85\times 0.85-0.85\times 0.15+0.15\times 0.15} \\ =\frac{{\left(0.85\right)}^3+{\left(0.15\right)}^3}{{\left(0.85\right)}^2-0.85\times 0.15+{\left(0.15\right)}^2} \\ \mathrm{We}\mathrm{know} \\ {a}^3+b^3=\left(a+b\right)\left(a^2+b^2-ab\right) \\ \mathrm{Here}a=0.85,b=0.15 \end{gathered}$$
$$\begin{gathered} \frac{{\left(0.85\right)}^3+{\left(0.15\right)}^3}{{\left(0.85\right)}^2-0.85\times 0.15+{\left(0.15\right)}^2} \\ =\frac{\left(0.85+0.15\right)\left({\left(0.85\right)}^2-0.85\times 0.15+{\left(0.15\right)}^2\right)}{{\left(0.85\right)}^2-0.85\times 0.15+{\left(0.15\right)}^2} \\ =0.85+0.15=1:\mathrm{RHS} \end{gathered}$$
Thus, LHS = RHS