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:

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