Упр.10.13 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) 2a^(1/2)(a^(1/2)-4)+8a^(1/2); 6) (b^0,4+3)^2-6b^0,4;
2) (a^0,5-3b^0,3)(2a^0,5+b^0,3); 7) (c^(1/3)-1)(c^(2/3)+c^(1/3)+1);
3) (3b^(2/3)-c^(3/2))(3b^(2/3)+c^(3/2)); 8) (a^(1/3)+a^(1/2))(a^(2/3)-a^(5/6)+a);
4) (a^(1/3)+b^(1/3))^2; 9) (a^(1/6)+b^(1/6))(a^(1/6)-b^(1/6))(a^(1/3)-b^(1/3));
5) (a^(-1/2)-(1/4)a^(-1/6))^2; 10) (x^(2/9)-1)(x^(4/9)+x^(2/9)+1)(x^(2/3)+1).

1. $$2a^{1/2}(a^{1/2}-4)+8a^{1/2}=2a^{1/2}\cdot a^{1/2}-8a^{1/2}+8a^{1/2}=2a.$$
2. $$\begin{aligned} (a^{0,5}-3b^{0,3})(2a^{0,5}+b^{0,3})&=2a^{0,5+0,5}+a^{0,5}b^{0,3}-6a^{0,5}b^{0,3}-3b^{0,3+0,3}\\ &=2a-5a^{0,5}b^{0,3}-3b^{0,6}\\ &=2a-5\sqrt{a}\sqrt[10]{b^3}-3\sqrt[5]{b^3}. \end{aligned}$$
3. $$\begin{aligned} (3b^{2/3}-c^{3/2})(3b^{2/3}+c^{3/2})&=(3b^{2/3})^2-(c^{3/2})^2\\ &=9b^{4/3}-c^3\\ &=9b\sqrt[3]{b}-c^3. \end{aligned}$$
4. $$\begin{aligned} (a^{1/3}+b^{1/3})^2&=a^{2/3}+2a^{1/3}b^{1/3}+b^{2/3}\\ &=\sqrt[3]{a^2}+2\sqrt[3]{ab}+\sqrt[3]{b^2}. \end{aligned}$$
5. $$\begin{aligned} \left(a^{-1/2}-\frac14 a^{-1/6}\right)^2 &=a^{-1}-\frac12 a^{-2/3}+\frac1{16}a^{-1/3}\\ &=\frac1a-\frac1{2\sqrt[3]{a^2}}+\frac1{16\sqrt[3]{a}}. \end{aligned}$$
6. $$\begin{aligned} (b^{0,4}+3)^2-6b^{0,4}&=b^{0,8}+6b^{0,4}+9-6b^{0,4}\\ &=b^{0,8}+9\\ &=b^{4/5}+9\\ &=\sqrt[5]{b^4}+9. \end{aligned}$$
7. $$\begin{aligned} (c^{1/3}-1)(c^{2/3}+c^{1/3}+1)&=c-1. \end{aligned}$$
8. $$\begin{aligned} (a^{1/3}+a^{1/2})(a^{2/3}-a^{5/6}+a) &=a^{1/3+2/3}+a^{1/2+2/3}-a^{1/3+5/6}-a^{1/2+5/6}+a^{4/3}+a^{3/2}\\ &=a+a^{7/6}-a^{7/6}-a^{4/3}+a^{4/3}+a^{3/2}\\ &=a+a^{1/2}\\ &=a+\sqrt{a}. \end{aligned}$$
9. $$\begin{aligned} (a^{1/6}+b^{1/6})(a^{1/6}-b^{1/6})(a^{1/3}-b^{1/3}) &=\left(a^{1/3}-b^{1/3}\right)\left(a^{1/3}-b^{1/3}\right)\\ &=a^{2/3}-2a^{1/3}b^{1/3}+b^{2/3}\\ &=\sqrt[3]{a^2}-2\sqrt[3]{ab}+\sqrt[3]{b^2}. \end{aligned}$$
10. $$\begin{aligned} (x^{2/9}-1)(x^{4/9}+x^{2/9}+1)(x^{2/3}+1) &=\left(x^{2/3}-1\right)\left(x^{2/3}+1\right)\\ &=x^{4/3}-1\\ &=\sqrt[3]{x^4}-1. \end{aligned}$$

Ответ

1) $$2a$$; 2) $$2a-5\sqrt{a}\sqrt[10]{b^3}-3\sqrt[5]{b^3}$$; 3) $$9b\sqrt[3]{b}-c^3$$; 4) $$\sqrt[3]{a^2}+2\sqrt[3]{ab}+\sqrt[3]{b^2}$$; 5) $$\frac1a-\frac1{2\sqrt[3]{a^2}}+\frac1{16\sqrt[3]{a}}$$; 6) $$\sqrt[5]{b^4}+9$$; 7) $$c-1$$; 8) $$a+\sqrt{a}$$; 9) $$\sqrt[3]{a^2}-2\sqrt[3]{ab}+\sqrt[3]{b^2}$$; 10) $$\sqrt[3]{x^4}-1$$.