Question

Возведите в степень выражение
$( \frac{1}{3} ab ^{2} + b ^{2}) ^{ 3}$

Answer 1

$\displaystyle ( \frac{1}{3}ab^{2}+b^{2})^{3}=( \frac{1}{3}ab^{2})^{3}+3( \frac{1}{3}ab^{2})^{2}b^{2}+3( \frac{1}{3}ab^{2})(b^{2})^{2}+(b^{2})^{3}= \\ \\ = \frac{a^{3}b^{6}}{27}+3b^{2}* \frac{a^{2}b^{4}}{9}+3b^{4}* \frac{ab^{2}}{3}+b^{6}= \frac{a^{3}b^{6}}{27}+ \frac{a^{2}b^{6}}{3}+ab^{6}+b^{6}= \\ \\ =b^{6}( \frac{a^{3}}{27}+ \frac{a^{2}}{3}+a+1)$

Или так:

$\displaystyle ( \frac{1}{3}ab^{2}+b^{2})^{3}=(b^{2}( \frac{a}{3}+1))^{3}=b^{6}( \frac{a}{3}+1)^{3}= \\ \\ =b^{6}(( \frac{a}{3})^{3}+3*( \frac{a}{3})^{2}+3* \frac{a}{3}+1)=b^{6}( \frac{a^{3}}{27}+ \frac{a^{2}}{3}+a+1)$