Question

Solve the equation:-
${(\frac{2}{3}{m}^2+\frac{5}{6}{n}^2)}^2$

Answer 1

${(\frac{2}{3}{m}^2+\frac{5}{6}{n}^2)}^2$

As we know that,

${(a+b)}^2=a^2+b^2+2ab$

Here,

$a=\frac{2}{3}{m}^2,b=\frac{5}{6}{n}^2$

Therefore,

${(\frac{2}{3}{m}^2+\frac{5}{6}{n}^2)}^2$

$={\left(\frac{2}{3}{m}^2\right)}^2+{\left(\frac{5}{6}{n}^2\right)}^2+2\left(\frac{2}{3}{m}^2\right)\left(\frac{5}{6}{n}^2\right)$

$=\frac{4}{9}{m}^4+\frac{25}{36}{n}^4+\frac{20}{18}{m}^2{n}^2$

$=\frac{4\times \left(4m^4\right)+1\times \left(25n^4\right)+2\left(20m^2{n}^2\right)}{36}$

$=\frac{16m^4+40m^2{n}^2+25n^4}{36}$

Hence ${(\frac{2}{3}{m}^2+\frac{5}{6}{n}^2)}^2=\frac{16m^4+40m^2{n}^2+25n^4}{36}$