Упр.37.30 ГДЗ Мордкович 10-11 класс (Алгебра)

1. Упростите выражение:
   а) $$\left(1+c^{1/2}\right)^2-2c^{1/2}$$;
   б) $$\left(m^{1/4}-m^{1/3}\right)^2+2m^{7/12}$$;
   в) $$\left(x^{1/2}-y^{1/2}\right)^2+2x^{1/2}y^{1/2}$$;
   г) $$\sqrt{b}+\sqrt{c}-\left(b^{1/4}+c^{1/4}\right)^2$$.

а) $$\left(1+c^{\frac12}\right)^2-2c^{\frac12}=1+2c^{\frac12}+c-2c^{\frac12}=1+c.$$

Ответ: $$1+c.$$

б) $$\left(m^{\frac14}-m^{\frac13}\right)^2+2m^{\frac7{12}}=m^{\frac12}-2m^{\frac14+\frac13}+m^{\frac23}+2m^{\frac7{12}}.$$

Так как $$\frac14+\frac13=\frac7{12},$$ то

$$m^{\frac12}-2m^{\frac7{12}}+m^{\frac23}+2m^{\frac7{12}}=m^{\frac12}+m^{\frac23}.$$

Ответ: $$m^{\frac12}+m^{\frac23}.$$

в) $$\left(x^{\frac12}-y^{\frac12}\right)^2+2x^{\frac12}y^{\frac12}=x-2x^{\frac12}y^{\frac12}+y+2x^{\frac12}y^{\frac12}=x+y.$$

Ответ: $$x+y.$$

г) $$\sqrt{b}+\sqrt{c}-\left(b^{\frac14}+c^{\frac14}\right)^2=\sqrt{b}+\sqrt{c}-\left(b^{\frac12}+2b^{\frac14}c^{\frac14}+c^{\frac12}\right).$$

Тогда

$$\sqrt{b}+\sqrt{c}-\sqrt{b}-2b^{\frac14}c^{\frac14}-\sqrt{c}=-2b^{\frac14}c^{\frac14}=-2\sqrt[4]{bc}.$$

Ответ: $$-2\sqrt[4]{bc}.$$