Упр.23.25 ГДЗ Мерзляк Поляков 9 класс (Алгебра)
1) (a-b)/(a^0,5-b^0,5)-(a^1,5-b^1,5)/(a-b);
2) (a^0,5-b^0,5)/(a^0,5+b^0,5)+(a^0,5+b^0,5)/(a^0,5-b^0,5);
3) (a^(1/2)+2a^(1/4) b^(1/4)+b^(1/2))/(a^(7/6) b^(5/6)-a^(5/6) b^(7/6))·(a-a^(1/3) b^(2/3))/(a^(1/4) b^(1/4)+b^(1/2));
4) (a^(3/2)+b^(3/2))/(a^2-ab)^(2/3):(a^(-2/3)(a-b)^(1/3))/(a^(3/2)-b^(3/2)).
$$\frac{a-b}{a^{0,5}-b^{0,5}}-\frac{a^{1,5}-b^{1,5}}{a-b}$$
$$=\frac{a-b}{a^{0,5}-b^{0,5}}-\frac{a^{1,5}-b^{1,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
$$=\frac{(a-b)(a^{0,5}+b^{0,5})-(a^{1,5}-b^{1,5})}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
$$=\frac{a^{1,5}+ab^{0,5}-ba^{0,5}-b^{1,5}-a^{1,5}+b^{1,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
$$=\frac{ab^{0,5}-ba^{0,5}}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
$$=\frac{a^{0,5}b^{0,5}(a^{0,5}-b^{0,5})}{(a^{0,5}-b^{0,5})(a^{0,5}+b^{0,5})}$$
$$=\frac{a^{0,5}b^{0,5}}{a^{0,5}+b^{0,5}}.$$$$\frac{a^{0,5}-b^{0,5}}{a^{0,5}+b^{0,5}}+\frac{a^{0,5}+b^{0,5}}{a^{0,5}-b^{0,5}}$$
$$=\frac{(a^{0,5}-b^{0,5})^2+(a^{0,5}+b^{0,5})^2}{(a^{0,5}+b^{0,5})(a^{0,5}-b^{0,5})}$$
$$=\frac{a-2a^{0,5}b^{0,5}+b+a+2a^{0,5}b^{0,5}+b}{a-b}$$
$$=\frac{2a+2b}{a-b}=\frac{2(a+b)}{a-b}.$$$$\frac{a^{1/2}+2a^{1/4}b^{1/4}+b^{1/2}}{a^{7/6}b^{5/6}-a^{5/6}b^{7/6}}\cdot\frac{a-a^{1/3}b^{2/3}}{a^{1/4}b^{1/4}+b^{1/2}}$$
$$=\frac{(a^{1/4}+b^{1/4})^2}{a^{5/6}b^{5/6}(a^{2/6}-b^{2/6})}\cdot\frac{a^{1/3}(a^{2/3}-b^{2/3})}{b^{1/4}(a^{1/4}+b^{1/4})}$$
$$=\frac{(a^{1/4}+b^{1/4})^2\cdot a^{1/3}(a^{2/3}-b^{2/3})}{a^{5/6}b^{5/6}(a^{1/3}-b^{1/3})(a^{1/4}+b^{1/4})\cdot b^{1/4}}$$
$$=\frac{(a^{1/4}+b^{1/4})\cdot a^{1/3}(a^{2/3}-b^{2/3})}{a^{5/6}b^{13/12}(a^{1/3}-b^{1/3})}.$$$$\frac{a^{3/2}+b^{3/2}}{(a^2-ab)^{2/3}}:\frac{a^{-2/3}(a-b)^{1/3}}{a^{3/2}-b^{3/2}}$$
$$=\frac{a^{3/2}+b^{3/2}}{(a(a-b))^{2/3}}\cdot\frac{a^{3/2}-b^{3/2}}{a^{-2/3}(a-b)^{1/3}}$$
$$=\frac{a^{3/2}+b^{3/2}}{a^{2/3}(a-b)^{2/3}}\cdot\frac{a^{3/2}-b^{3/2}}{a^{-2/3}(a-b)^{1/3}}$$
$$=\frac{(a^{3/2}+b^{3/2})(a^{3/2}-b^{3/2})}{(a-b)}$$
$$=\frac{a^3-b^3}{a-b}=\frac{(a-b)(a^2+ab+b^2)}{a-b}=a^2+ab+b^2.$$
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) $$a^2+ab+b^2$$
