Упр.37.39 ГДЗ Мордкович 10-11 класс (Алгебра)

1. а) $$\frac{a^{3/2}-b^{3/2}}{a^{1/2}+b^{1/2}}\cdot\frac{a-b}{a+a^{1/2}b^{1/2}+b}+2a^{1/2}b^{1/2}$$

а)

$$\frac{a^{\frac32}-b^{\frac32}}{a^{\frac12}+b^{\frac12}}\cdot \frac{a-b}{a+a^{\frac12}b^{\frac12}+b}+2a^{\frac12}b^{\frac12}$$

Представим разности как произведения:
$$a^{\frac32}-b^{\frac32}=(a^{\frac12}-b^{\frac12})(a+a^{\frac12}b^{\frac12}+b),$$
$$a-b=(a^{\frac12}-b^{\frac12})(a^{\frac12}+b^{\frac12}).$$
Тогда
$$\frac{(a^{\frac12}-b^{\frac12})(a+a^{\frac12}b^{\frac12}+b)}{a^{\frac12}+b^{\frac12}} \cdot \frac{(a^{\frac12}-b^{\frac12})(a^{\frac12}+b^{\frac12})}{a+a^{\frac12}b^{\frac12}+b} +2a^{\frac12}b^{\frac12}$$
$$=(a^{\frac12}-b^{\frac12})^2+2a^{\frac12}b^{\frac12}.$$
Раскроем квадрат:
$$(a^{\frac12}-b^{\frac12})^2+2a^{\frac12}b^{\frac12} =a-2a^{\frac12}b^{\frac12}+b+2a^{\frac12}b^{\frac12}=a+b.$$

б)

$$\left(\frac{q^{\frac12}}{p-p^{\frac12}q^{\frac12}}+\frac{p^{\frac12}}{q-p^{\frac12}q^{\frac12}}\right)\cdot \frac{pq^{\frac12}+p^{\frac12}q}{p-q}$$

Вынесем в знаменателях:
$$p-p^{\frac12}q^{\frac12}=p^{\frac12}\left(p^{\frac12}-q^{\frac12}\right),\qquad q-p^{\frac12}q^{\frac12}=q^{\frac12}\left(q^{\frac12}-p^{\frac12}\right).$$
Тогда
$$\frac{q^{\frac12}}{p^{\frac12}(p^{\frac12}-q^{\frac12})} +\frac{p^{\frac12}}{q^{\frac12}(q^{\frac12}-p^{\frac12})} = \frac{q^{\frac12}}{p^{\frac12}(p^{\frac12}-q^{\frac12})} -\frac{p^{\frac12}}{q^{\frac12}(p^{\frac12}-q^{\frac12})}.$$
Приведём к общему знаменателю:
$$\frac{q-p}{p^{\frac12}q^{\frac12}(p^{\frac12}-q^{\frac12})}.$$
Кроме того,
$$pq^{\frac12}+p^{\frac12}q=p^{\frac12}q^{\frac12}\left(p^{\frac12}+q^{\frac12}\right),$$
$$p-q=(p^{\frac12}-q^{\frac12})(p^{\frac12}+q^{\frac12}).$$
Следовательно,
$$\frac{q-p}{p^{\frac12}q^{\frac12}(p^{\frac12}-q^{\frac12})}\cdot \frac{p^{\frac12}q^{\frac12}(p^{\frac12}+q^{\frac12})}{(p^{\frac12}-q^{\frac12})(p^{\frac12}+q^{\frac12})} = \frac{q-p}{(p^{\frac12}-q^{\frac12})^2}.$$
Так как
$$q-p=-(p-q)=-(p^{\frac12}-q^{\frac12})(p^{\frac12}+q^{\frac12}),$$
то
$$\frac{q-p}{(p^{\frac12}-q^{\frac12})^2} = \frac{q^{\frac12}+p^{\frac12}}{q^{\frac12}-p^{\frac12}}.$$

Ответ: а) $$a+b$$; б) $$\frac{\sqrt q+\sqrt p}{\sqrt q-\sqrt p}$$.