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

Задача

а) (a^3/2 — b^3/2)/(a^1/2 + b^1/2) * (a — b)/(a + a^1/2 * b^1/2 + b) + 2a^1/2 * b^1/2;

Подробный ответ

а) Преобразуем выражение:

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

Разложим разности на множители:

$$ a^{3/2}-b^{3/2}=(a^{1/2}-b^{1/2})(a+b^{1/2}a^{1/2}+b^{1/2}) $$
$$ a-b=(a^{1/2}-b^{1/2})(a^{1/2}+b^{1/2}) $$

Тогда

$$ \frac{(a^{1/2}-b^{1/2})(a+a^{1/2}b^{1/2}+b)\cdot (a^{1/2}-b^{1/2})(a^{1/2}+b^{1/2})}{(a^{1/2}+b^{1/2})(a+a^{1/2}b^{1/2}+b)}+2a^{1/2}b^{1/2} $$

Сокращаем и получаем:

$$ (a^{1/2}-b^{1/2})^2+2a^{1/2}b^{1/2} $$

$$ =a-2a^{1/2}b^{1/2}+b+2a^{1/2}b^{1/2}=a+b $$

б) Упростим выражение:

$$ \left(\frac{q^{1/2}}{p-p^{1/2}q^{1/2}}+\frac{p^{1/2}}{q-p^{1/2}q^{1/2}}\right)\cdot \frac{pq^{1/2}+p^{1/2}q}{p-q} $$

Вынесем в знаменателях и числителе общие множители:

$$ p-p^{1/2}q^{1/2}=p^{1/2}(p^{1/2}-q^{1/2}), \qquad q-p^{1/2}q^{1/2}=q^{1/2}(q^{1/2}-p^{1/2}) $$
$$ pq^{1/2}+p^{1/2}q=p^{1/2}q^{1/2}(p^{1/2}+q^{1/2}), \qquad p-q=(p^{1/2}-q^{1/2})(p^{1/2}+q^{1/2}) $$

Тогда

$$ \left(\frac{q^{1/2}}{p^{1/2}(p^{1/2}-q^{1/2})}+\frac{p^{1/2}}{q^{1/2}(q^{1/2}-p^{1/2})}\right)\cdot \frac{p^{1/2}q^{1/2}(p^{1/2}+q^{1/2})}{(p^{1/2}-q^{1/2})(p^{1/2}+q^{1/2})} $$

После приведения к общему знаменателю и сокращения получаем:

$$ \frac{q-p}{(p^{1/2}-q^{1/2})^2}\cdot \frac{1}{p^{1/2}q^{1/2}} = \frac{(q^{1/2}-p^{1/2})(q^{1/2}+p^{1/2})}{(q^{1/2}-p^{1/2})^2} $$

$$ =\frac{q^{1/2}+p^{1/2}}{q^{1/2}-p^{1/2}} $$