Question

Find the following squares by using the identities${\left[\left(\frac{2m}{3}\right)+\left(\frac{3n}{2}\right)\right]}^2$.

Answer 1

Evaluate the given expression

Given, ${\left[\left(\frac{2m}{3}\right)+\left(\frac{3n}{2}\right)\right]}^2$

We use the identity, ${(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}$

$$\begin{gathered} {\left[\left(\frac{2m}{3}\right)+\left(\frac{3n}{2}\right)\right]}^2 \\ ={\left(\frac{2m}{3}\right)}^2+{\left(\frac{3n}{2}\right)}^2+2\left(\frac{2m}{3}\right)\left(\frac{3n}{2}\right) \\ =\frac{4m^2}{9}+\frac{9n^2}{4}+2mn \end{gathered}$$

Hence, the required answer is $\frac{4m^2}{9}+\frac{9n^2}{4}+2mn$.