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

1) (a^(v5)+2)(a^(v5)-2)-(a^(v5)+3)^2;
2) (a^(2v7)-a^(v7))/(a^(4v7)-a^(3v7);
3) (a^(2v3)-b^(2v2))/(a^(v3)+b^(v2))^2+1;
4) (a^(24^(1/3))-1)/(a^(3^(1/3))-1)-(a^(81^(1/3))+1)/(a^(3^(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) $$2a^{\sqrt[3]{3}}-a^{2\sqrt[3]{3}}$$.