Упр.1.15 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1. Упростите выражение:

   1) $$(a^{\sqrt{5}}+2)(a^{\sqrt{5}}-2)-(a^{\sqrt{5}}+3)^2$$;

   2) $$\frac{a^{2\sqrt{7}}-a^{\sqrt{7}}}{a^{4\sqrt{7}}-a^{3\sqrt{7}}}$$;

   3) $$\frac{a^{2\sqrt{3}}-b^{2\sqrt{2}}}{(a^{\sqrt{3}}+b^{\sqrt{2}})^2}+1$$;

   4) $$\frac{a^{24^{\frac{1}{3}}}-1}{a^{3^{\frac{1}{3}}}-1}-\frac{a^{81^{\frac{1}{3}}}+1}{a^{3^{\frac{1}{3}}}+1}$$.

1. $$\left(a^{\sqrt5}+2\right)\left(a^{\sqrt5}-2\right)-\left(a^{\sqrt5}+3\right)^2$$
   $$=a^{2\sqrt5}-4-\left(a^{2\sqrt5}+6a^{\sqrt5}+9\right)$$
   $$=-6a^{\sqrt5}-13.$$

2. $$\frac{a^{2\sqrt7}-a^{\sqrt7}}{a^{4\sqrt7}-a^{3\sqrt7}}= \frac{a^{\sqrt7}\left(a^{\sqrt7}-1\right)}{a^{3\sqrt7}\left(a^{\sqrt7}-1\right)}= a^{-2\sqrt7}=\frac{1}{a^{2\sqrt7}}.$$

3. $$\frac{a^{2\sqrt3}-b^{2\sqrt2}}{\left(a^{\sqrt3}+b^{\sqrt2}\right)^2}+1= \frac{\left(a^{\sqrt3}-b^{\sqrt2}\right)\left(a^{\sqrt3}+b^{\sqrt2}\right)}{\left(a^{\sqrt3}+b^{\sqrt2}\right)^2}+1$$
   $$= \frac{a^{\sqrt3}-b^{\sqrt2}}{a^{\sqrt3}+b^{\sqrt2}}+1= \frac{a^{\sqrt3}-b^{\sqrt2}+a^{\sqrt3}+b^{\sqrt2}}{a^{\sqrt3}+b^{\sqrt2}}= \frac{2a^{\sqrt3}}{a^{\sqrt3}+b^{\sqrt2}}.$$

4. $$\frac{a^{\sqrt[3]{24}}-1}{a^{\sqrt[3]{3}}-1}-\frac{a^{\sqrt[3]{81}}+1}{a^{\sqrt[3]{3}}+1}= \frac{a^{2\sqrt[3]{3}}-1}{a^{\sqrt[3]{3}}-1}-\frac{a^{3\sqrt[3]{3}}+1}{a^{\sqrt[3]{3}}+1}$$
   $$= \frac{\left(a^{\sqrt[3]{3}}-1\right)\left(a^{\sqrt[3]{3}}+1\right)}{a^{\sqrt[3]{3}}-1} -\frac{\left(a^{\sqrt[3]{3}}+1\right)\left(a^{2\sqrt[3]{3}}-a^{\sqrt[3]{3}}+1\right)}{a^{\sqrt[3]{3}}+1}$$
   $$= \left(a^{\sqrt[3]{3}}+1\right)-\left(a^{2\sqrt[3]{3}}-a^{\sqrt[3]{3}}+1\right) =2a^{\sqrt[3]{3}}-a^{2\sqrt[3]{3}}.$$

Ответ

1) $$-6a^{\sqrt5}-13$$;
2) $$\frac{1}{a^{2\sqrt7}}$$;
3) $$\frac{2a^{\sqrt3}}{a^{\sqrt3}+b^{\sqrt2}}$$;
4) $$2a^{\sqrt[3]{3}}-a^{2\sqrt[3]{3}}$$.