Question

Упростите выражения
Даю 30 баллов

Answer 1

$\displaystyle\bf\\11)\\\\1) \ \ \frac{1}{(x-1)^{2} } -\frac{x}{1-x^{2} } = \frac{1}{(1-x)^{2} } -\frac{x}{(1-x)(1+x) } =\\\\\\=\frac{1+x-x\cdot(1-x)}{(1-x)^{2} \cdot(1+x)} =\frac{1+x-x+x^{2} }{(1-x)^{2} \cdot(1+x)} =\frac{1+x^{2} }{(1-x)^{2} \cdot(1+x)} \\\\\\2) \ \ \frac{1-x^{2} }{1+x^{2} } \cdot\frac{1+x^{2} }{(1-x)^{2} \cdot(1+x)} = \frac{(1-x)(1+x) }{1+x^{2} } \cdot\frac{1+x^{2} }{(1-x)^{2} \cdot(1+x)} =\\\\\\=\frac{1}{1-x} \\\\\\3) \ \ \frac{x}{1-x} -\frac{1}{1-x} =\frac{x-1}{1-x} =-1$

$\displaystyle\bf\\12)\\\\\frac{a^{2}+ab+b^{2} }{a^{3}- b^{3} } -\frac{a^{2} -ab+b^{2} }{a^{3} + b^{3} } =\\\\\\=\frac{a^{2}+ab+b^{2} }{(a- b)\cdot(a^{2} +ab+b^{2} ) } -\frac{a^{2} -ab+b^{2} }{(a+ b)\cdot(a^{2} -ab+b^{2} ) } =\\\\\\=\frac{1}{a-b} -\frac{1}{a+b} =\frac{a+b-a+b}{(a-b)(a+b)} =\frac{2b}{a^{2}-b^{2} }$