Самостоятельная работа 18 Вариант 2 ГДЗ Дидактические материалы Никольский Потапов 10 класс (Алгебра)

2. Упростите, применив формулы сокращенного умножения:
а) (m^(1/2)+n)^2+(m^(1/2)-n)^2; б) (m^(1/4)-2n^(1/3))-(m^(1/4)+2n^(1/3))^2;
в) (m^(1/2)-n^(1/4))(m^(1/2)+n^(1/4)); г) (m^(1/2)-n)(m+m^(1/2) n+n^2).
3. Вычислите (6^(1/2)+2^(1/2))^2+(6^(1/2)-2^(1/2))^2.
4. Сократите дробь (x-y)/(x^(1/2)+y^(1/2)).
5. Упростите ((x^(1/4)+y^(1/4))/(x^(1/4)-y^(1/4))-(x^(1/4)-y^(1/4))/(x^(1/4)+y^(1/4)))·(y^(-1/2)-x^(-1/2)).

1. Запишем корни в виде степеней:

   $$\sqrt{5}=5^{\frac12}, \qquad \sqrt[3]{4}=4^{\frac13}, \qquad \sqrt[5]{2^6}=(2^6)^{\frac15}=2^{\frac65}.$$

2. Упростим выражения, используя формулы сокращённого умножения:

   а)

   $$\left(m^{\frac12}+n\right)^2+\left(m^{\frac12}-n\right)^2$$

   $$= \left(m+2m^{\frac12}n+n^2\right)+\left(m-2m^{\frac12}n+n^2\right)=2m+2n^2.$$

   б)

   $$\left(m^{\frac14}-2n^{\frac13}\right)^2-\left(m^{\frac14}+2n^{\frac13}\right)^2$$

   $$=\left[\left(m^{\frac14}-2n^{\frac13}\right)-\left(m^{\frac14}+2n^{\frac13}\right)\right]\left[\left(m^{\frac14}-2n^{\frac13}\right)+\left(m^{\frac14}+2n^{\frac13}\right)\right]$$

   $$=(-4n^{\frac13})(2m^{\frac14})=-8m^{\frac14}n^{\frac13}.$$

   в)

   $$\left(m^{\frac12}-n^{\frac14}\right)\left(m^{\frac12}+n^{\frac14}\right)=\left(m^{\frac12}\right)^2-\left(n^{\frac14}\right)^2=m-n^{\frac12}.$$

   г)

   $$\left(m^{\frac12}-n\right)\left(m+m^{\frac12}n+n^2\right)=m^{\frac32}-n^3.$$

3. Вычислим значение выражения:

   $$\left(6^{\frac12}+2^{\frac12}\right)^2+\left(6^{\frac12}-2^{\frac12}\right)^2$$

   $$=2\left(6+2\right)=16.$$

4. Сократим дробь:

   $$\frac{x-y}{x^{\frac12}+y^{\frac12}}=\frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{\sqrt{x}+\sqrt{y}}=\sqrt{x}-\sqrt{y}.$$

5. Упростим выражение:

   $$\left(\frac{x^{\frac14}+y^{\frac14}}{x^{\frac14}-y^{\frac14}}-\frac{x^{\frac14}-y^{\frac14}}{x^{\frac14}+y^{\frac14}}\right)\left(y^{-\frac12}-x^{-\frac12}\right)$$

   $$=\frac{\left(x^{\frac14}+y^{\frac14}\right)^2-\left(x^{\frac14}-y^{\frac14}\right)^2}{\left(x^{\frac14}-y^{\frac14}\right)\left(x^{\frac14}+y^{\frac14}\right)}\left(\frac1{\sqrt{y}}-\frac1{\sqrt{x}}\right)$$

   $$=\frac{4x^{\frac14}y^{\frac14}}{x^{\frac12}-y^{\frac12}}\cdot\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}\sqrt{y}}=\frac{4}{x^{\frac14}y^{\frac14}}.$$

Ответ

1) $$5^{\frac12},\ 4^{\frac13},\ 2^{\frac65}$$

2) а) $$2m+2n^2$$; б) $$-8m^{\frac14}n^{\frac13}$$; в) $$m-n^{\frac12}$$; г) $$m^{\frac32}-n^3$$

3) $$16$$

4) $$\sqrt{x}-\sqrt{y}$$

5) $$\frac{4}{x^{\frac14}y^{\frac14}}$$