Question

100 баллов!

Выполни вычитание:

$\frac{1}{u-w} - \frac{3uw}{u^{3}-w^{3} }$

Answer 1

Ответ:

Объяснение:

$a^{3} -b^{3} =(a-b)(a^{2}+ab+b^{2} )$

$\frac{1}{u-w} -\frac{3uw}{u^{3}-w^{3} } =\frac{1}{u-w} -\frac{3uw}{(u-w)(u^{2} +uw+w^{2} )}=\frac{u^{2} +uw+w^{2} -3uw}{(u-w)(u^{2} +uw+w^{2} )} =\frac{u^{2} -2uw+w^{2} }{(u-w)(u^{2} +uw+w^{2} )}=\\=\frac{(u-w)^{2} }{(u-w)(u^{2} +uw+w^{2} )} =\frac{u-w}{u^{2}+uw+w^{2} }$

Answer 2

Ответ:

$\large\boldsymbol{ \dfrac{u- w}{u^2 + u w + w^2 } }$

Объяснение:

Выполни вычитание:

$\large \boldsymbol{} \displaystyle \frac{1}{u - w} - \frac{3uw}{u^3 - w^3}$

Вспомним :


$\large \boldsymbol{a^3 - b^3 = (a-b)(a^2 + ab + b^2)}$

Тогда

$\large \boldsymbol{} \displaystyle \frac{1}{u - w} - \frac{3uw}{u^3 - w^3} = \frac{1}{u - w} - \frac{3uw}{(u-w)(u^2 + u w+ w^2)}$

Приведем дроби  к общему знаменателю


$\large \boldsymbol{} \displaystyle \frac{1}{u - w} \cdot \frac{(u^2 + u w+ w^2)}{(u^2 + u w+ w^2)} - \frac{3uw}{(u-w)(u^2 + u w+ w^2)} = \\\\\\\\ = \frac{u ^2 + u w^2 + w^2 - 3uw }{(u-w)(u^2 + u w+ w^2)} = \frac{u^2 - 2uw + w^2}{(u-w)(u^2 + u w+ w^2)} = \\\\\\\\\ = \frac{(u - w)^{ \slash \!\!\!2}}{\Big \slash \!\!\!\!\!\!\!\!\!\!\!(u-w)(u^2 + u w+ w^2)} =\boxed{ \frac{u- w}{u^2 + u w + w^2 } }$