Упр.9.29 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) (va-vb)/(a^(1/4)+b^(1/4));
2) (x^(1/6)-9)/(x^(1/12)+3);
3) (vm+m^(1/4))/(m-(m^3)^(1/4));
4) ((ab^2)^(1/8)-(a^2 b)^(1/8))/(a^(1/4)-b^(1/4));
5) (a(b^2)^(1/3)-b(a^2)^(1/3))/(a^2 b^2)^(1/3);
6) ((x^2)^(1/3)+4(x^(1/3))+16)/(x-64);
7) (va+vb)/(a^(1/3)-(ab)^(1/6)+b^(1/3));
8) (2-2^(1/3))/2^(1/3);
9) ((a^3)^(1/4)-a^(1/4)+va-1)/(a-va).

1. $$\frac{\sqrt a-\sqrt b}{\sqrt[4]{a}+\sqrt[4]{b}}= \frac{(\sqrt[4]{a})^2-(\sqrt[4]{b})^2}{\sqrt[4]{a}+\sqrt[4]{b}}= \frac{(\sqrt[4]{a}-\sqrt[4]{b})(\sqrt[4]{a}+\sqrt[4]{b})}{\sqrt[4]{a}+\sqrt[4]{b}}= \sqrt[4]{a}-\sqrt[4]{b}.$$

2. $$\frac{\sqrt[6]{x}-9}{\sqrt[12]{x}+3}= \frac{(\sqrt[12]{x})^2-3^2}{\sqrt[12]{x}+3}= \frac{(\sqrt[12]{x}-3)(\sqrt[12]{x}+3)}{\sqrt[12]{x}+3}= \sqrt[12]{x}-3.$$

3. $$\frac{\sqrt m+\sqrt[4]{m}}{m-\sqrt[4]{m^3}}= \frac{(\sqrt[4]{m})^2+\sqrt[4]{m}}{(\sqrt[4]{m})^4-(\sqrt[4]{m})^3}= \frac{\sqrt[4]{m}(\sqrt[4]{m}+1)}{(\sqrt[4]{m})^3(\sqrt[4]{m}-1)}.$$

   Сократим на $$\sqrt[4]{m}$$:
   $$\frac{\sqrt[4]{m}+1}{\sqrt[4]{m^3}-\sqrt m}.$$

4. $$\frac{\sqrt[8]{ab^2}-\sqrt[8]{a^2b}}{\sqrt[4]{a}-\sqrt[4]{b}}= \frac{\sqrt[8]{ab}\,(\sqrt[8]{b}-\sqrt[8]{a})}{(\sqrt[8]{a})^2-(\sqrt[8]{b})^2}.$$

   Тогда
   $$\frac{\sqrt[8]{ab}\,(\sqrt[8]{b}-\sqrt[8]{a})}{(\sqrt[8]{a}-\sqrt[8]{b})(\sqrt[8]{a}+\sqrt[8]{b})} =-\frac{\sqrt[8]{ab}}{\sqrt[8]{a}+\sqrt[8]{b}}.$$

5. $$\frac{a\sqrt[3]{b^2}-b\sqrt[3]{a^2}}{\sqrt[3]{a^2b^2}}= \frac{\sqrt[3]{a^3b^2}-\sqrt[3]{b^3a^2}}{\sqrt[3]{a^2b^2}}= \frac{\sqrt[3]{a^2b^2}\,(\sqrt[3]{a}-\sqrt[3]{b})}{\sqrt[3]{a^2b^2}}.$$

   Получаем
   $$\sqrt[3]{a}-\sqrt[3]{b}.$$

6. $$\frac{\sqrt[3]{x^2}+4\sqrt[3]{x}+16}{x-64}= \frac{(\sqrt[3]{x})^2+4\sqrt[3]{x}+4^2}{(\sqrt[3]{x})^3-4^3}.$$

   По формуле суммы кубов:
   $$\frac{(\sqrt[3]{x}+4)(\sqrt[3]{x}-4)}{(\sqrt[3]{x}-4)(x+4\sqrt[3]{x}+16)}= \frac{1}{\sqrt[3]{x}-4}.$$

7. $$\frac{\sqrt a+\sqrt b}{\sqrt[3]{a}-\sqrt[6]{ab}+\sqrt[3]{b}}= \frac{\sqrt a+\sqrt b}{(\sqrt[6]{a})^2-\sqrt[6]{a}\sqrt[6]{b}+(\sqrt[6]{b})^2}.$$

   Умножим числитель и знаменатель на $$\sqrt[6]{a}+\sqrt[6]{b}$$:
   $$\frac{(\sqrt a+\sqrt b)(\sqrt[6]{a}+\sqrt[6]{b})}{(\sqrt[6]{a})^3+(\sqrt[6]{b})^3}= \frac{(\sqrt a+\sqrt b)(\sqrt[6]{a}+\sqrt[6]{b})}{\sqrt a+\sqrt b}= \sqrt[6]{a}+\sqrt[6]{b}.$$

8. $$\frac{2-2^{1/3}}{2^{1/3}}= \frac{2^{1/3}\cdot 2^{2/3}-2^{1/3}}{2^{1/3}}= 2^{2/3}-1=\sqrt[3]{4}-1.$$

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

   Вынесем общий множитель:
   $$\frac{(\sqrt[4]{a}+1)(\sqrt a-1)}{\sqrt a(\sqrt a-1)}= \frac{\sqrt[4]{a}+1}{\sqrt a}.$$

Ответ

1. $$\sqrt[4]{a}-\sqrt[4]{b}$$
2. $$\sqrt[12]{x}-3$$
3. $$\frac{\sqrt[4]{m}+1}{\sqrt[4]{m^3}-\sqrt m}$$
4. $$-\frac{\sqrt[8]{ab}}{\sqrt[8]{a}+\sqrt[8]{b}}$$
5. $$\sqrt[3]{a}-\sqrt[3]{b}$$
6. $$\frac{1}{\sqrt[3]{x}-4}$$
7. $$\sqrt[6]{a}+\sqrt[6]{b}$$
8. $$\sqrt[3]{4}-1$$
9. $$\frac{\sqrt[4]{a}+1}{\sqrt a}$$