Упр.23.25 ГДЗ Мерзляк Поляков 9 класс (Алгебра)

Упростите выражение:

1. $$\frac{a-b}{a^{0,5}-b^{0,5}}-\frac{a^{1,5}-b^{1,5}}{a-b}$$;
2. $$\frac{a^{0,5}-b^{0,5}}{a^{0,5}+b^{0,5}}+\frac{a^{0,5}+b^{0,5}}{a^{0,5}-b^{0,5}}$$;
3. $$\frac{a^{1/2}+2a^{1/4}b^{1/4}+b^{1/2}}{a^{7/6}b^{5/6}-a^{5/6}b^{7/6}}\cdot\frac{a-a^{1/3}b^{2/3}}{a^{1/4}b^{1/4}+b^{1/2}}$$;
4. $$\frac{a^{3/2}+b^{3/2}}{(a^2-ab)^{2/3}}:\frac{a^{-2/3}(a-b)^{1/3}}{a^{3/2}-b^{3/2}}$$.

1. $$\frac{a-b}{a^{0,5}-b^{0,5}}-\frac{a^{1,5}-b^{1,5}}{a-b}$$
   $$=\frac{a-b}{a^{0,5}-b^{0,5}}-\frac{a^{1,5}-b^{1,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
   $$=\frac{(a-b)(a^{0,5}+b^{0,5})-(a^{1,5}-b^{1,5})}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
   $$=\frac{a^{1,5}+ab^{0,5}-ba^{0,5}-b^{1,5}-a^{1,5}+b^{1,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
   $$=\frac{ab^{0,5}-ba^{0,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
   $$=\frac{a^{0,5}b^{0,5}(a^{0,5}-b^{0,5})}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
   $$=\frac{a^{0,5}b^{0,5}}{a^{0,5}+b^{0,5}}.$$

2. $$\frac{a^{0,5}-b^{0,5}}{a^{0,5}+b^{0,5}}+\frac{a^{0,5}+b^{0,5}}{a^{0,5}-b^{0,5}}$$
   $$=\frac{(a^{0,5}-b^{0,5})^2+(a^{0,5}+b^{0,5})^2}{(a^{0,5}+b^{0,5})(a^{0,5}-b^{0,5})}$$
   $$=\frac{a-2a^{0,5}b^{0,5}+b+a+2a^{0,5}b^{0,5}+b}{a-b}$$
   $$=\frac{2a+2b}{a-b}=\frac{2(a+b)}{a-b}.$$

3. $$\frac{a^{1/2}+2a^{1/4}b^{1/4}+b^{1/2}}{a^{7/6}b^{5/6}-a^{5/6}b^{7/6}}\cdot\frac{a-a^{1/3}b^{2/3}}{a^{1/4}b^{1/4}+b^{1/2}}$$
   $$=\frac{(a^{1/4}+b^{1/4})^2}{a^{5/6}b^{5/6}(a^{2/6}-b^{2/6})}\cdot\frac{a^{1/3}(a^{2/3}-b^{2/3})}{b^{1/4}(a^{1/4}+b^{1/4})}$$
   $$=\frac{(a^{1/4}+b^{1/4})^2\cdot a^{1/3}(a^{2/3}-b^{2/3})}{a^{5/6}b^{5/6}(a^{1/3}-b^{1/3})(a^{1/4}+b^{1/4})\cdot b^{1/4}}$$
   $$=\frac{(a^{1/4}+b^{1/4})\cdot a^{1/3}(a^{2/3}-b^{2/3})}{a^{5/6}b^{13/12}(a^{1/3}-b^{1/3})}.$$

4. $$\frac{a^{3/2}+b^{3/2}}{(a^2-ab)^{2/3}}:\frac{a^{-2/3}(a-b)^{1/3}}{a^{3/2}-b^{3/2}}$$
   $$=\frac{a^{3/2}+b^{3/2}}{(a(a-b))^{2/3}}\cdot\frac{a^{3/2}-b^{3/2}}{a^{-2/3}(a-b)^{1/3}}$$
   $$=\frac{a^{3/2}+b^{3/2}}{a^{2/3}(a-b)^{2/3}}\cdot\frac{a^{3/2}-b^{3/2}}{a^{-2/3}(a-b)^{1/3}}$$
   $$=\frac{(a^{3/2}+b^{3/2})(a^{3/2}-b^{3/2})}{(a-b)}$$
   $$=\frac{a^3-b^3}{a-b}=\frac{(a-b)(a^2+ab+b^2)}{a-b}=a^2+ab+b^2.$$

Ответ

1) $$\frac{a^{0,5}b^{0,5}}{a^{0,5}+b^{0,5}}$$
2) $$\frac{2(a+b)}{a-b}$$
3) $$\frac{(a^{1/4}+b^{1/4})a^{1/3}(a^{2/3}-b^{2/3})}{a^{5/6}b^{13/12}(a^{1/3}-b^{1/3})}$$
4) $$a^2+ab+b^2$$