Упр.450 ГДЗ Колмогоров 10-11 класс (Алгебра)
а) $$\frac{a^{2\sqrt2}-b^{2\sqrt3}}{(a^{\sqrt2}-b^{\sqrt3})^2}+1=\frac{(a^{\sqrt2})^2-(b^{\sqrt3})^2}{(a^{\sqrt2}-b^{\sqrt3})^2}+1$$
$$=\frac{(a^{\sqrt2}-b^{\sqrt3})(a^{\sqrt2}+b^{\sqrt3})}{(a^{\sqrt2}-b^{\sqrt3})^2}+1=\frac{a^{\sqrt2}+b^{\sqrt3}}{a^{\sqrt2}-b^{\sqrt3}}+1$$
$$=\frac{a^{\sqrt2}+b^{\sqrt3}+a^{\sqrt2}-b^{\sqrt3}}{a^{\sqrt2}-b^{\sqrt3}}=\frac{2a^{\sqrt2}}{a^{\sqrt2}-b^{\sqrt3}}.$$
б) $$\frac{(a^{2\sqrt3}-1)(a^{2\sqrt3}+a^{\sqrt3}+a^{3\sqrt3})}{a^{4\sqrt3}-a^{\sqrt3}}$$
$$=\frac{(a^{2\sqrt3}-1)\,a^{\sqrt3}(a^{\sqrt3}+1)(a^{2\sqrt3}+a^{\sqrt3}+1)}{a^{\sqrt3}(a^{3\sqrt3}-1)}$$
$$=\frac{(a^{\sqrt3}-1)(a^{\sqrt3}+1)(a^{2\sqrt3}+a^{\sqrt3}+1)}{a^{3\sqrt3}-1}$$
$$=\frac{a^{3\sqrt3}-1}{a^{3\sqrt3}-1}=a^{\sqrt3}+1.$$
в) $$\frac{a^{\sqrt5}-b^{\sqrt7}}{\frac{2\sqrt5}{3}a+\frac{\sqrt5}{3}a+\frac{\sqrt7}{3}b+\frac{2\sqrt7}{3}b}$$
$$=\frac{a^{\sqrt5}-b^{\sqrt7}}{\frac{\sqrt5}{3}a\left(2+1\right)+\frac{\sqrt7}{3}b\left(1+2\right)}$$
$$=\frac{a^{\sqrt5}-b^{\sqrt7}}{\sqrt5\,a+\sqrt7\,b}$$
$$=a^{\frac{\sqrt5}{3}}-b^{\frac{\sqrt7}{3}}.$$
г) $$\sqrt{(x^\pi+y^\pi)^2-\left(\frac{1}{4^\pi}xy\right)^\pi}$$
$$=\sqrt{x^{2\pi}+2x^\pi y^\pi+y^{2\pi}-4x^\pi y^\pi}$$
$$=\sqrt{x^{2\pi}-2x^\pi y^\pi+y^{2\pi}}=\sqrt{(x^\pi-y^\pi)^2}=|x^\pi-y^\pi|.$$
Ответ
а) $$\frac{2a^{\sqrt2}}{a^{\sqrt2}-b^{\sqrt3}}$$; б) $$a^{\sqrt3}+1$$; в) $$a^{\frac{\sqrt5}{3}}-b^{\frac{\sqrt7}{3}}$$; г) $$|x^\pi-y^\pi|$$.
