Question

Find the product ab(a²+b²) and evaluate it for a = 2 and b = $\frac{1}{2}$.

Answer 1

$$\begin{gathered} \mathrm{ab}{(}^2{+b}^2) \\ =\mathrm{a}\mathrm{b}\times {\mathrm{a}}^2+\mathrm{a}\mathrm{b}\times {\mathrm{b}}^2 \\ =\mathrm{a}\times {\mathrm{a}}^2\times \mathrm{b}+\mathrm{a}\times \mathrm{b}\times {\mathrm{b}}^2 \\ ={\mathrm{a}}^{(1+2)}\times \mathrm{b}+\mathrm{a}\times {\mathrm{b}}^{(1+2)} \\ ={\mathrm{a}}^3\mathrm{b}+\mathrm{a}{\mathrm{b}}^3 \\ \mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{a}=2\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{b}=\frac{1}{2},\mathrm{w}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{t}: \\ \mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{a}\mathrm{b}({\mathrm{a}}^2+{\mathrm{b}}^2)=2\times \frac{1}{2}(2^2+\frac{1}{2^2})=4+\frac{1}{4}=\frac{17}{4} \\ \mathrm{R}.\mathrm{H}.\mathrm{S}.={\mathrm{a}}^3\mathrm{b}+\mathrm{a}{\mathrm{b}}^3=2^3\times \frac{1}{2}+2\times {\left(\frac{1}{2}\right)}^3=4+\frac{1}{4}=\frac{17}{4} \\ \therefore \mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{R}.\mathrm{H}.\mathrm{S}. \end{gathered}$$