301. Преобразуйте выражение:

Решение:

1. \( \sqrt{\frac{p^8}{a^6}} = \frac{\sqrt{p^8}}{\sqrt{a^6}} = \frac{p^{8/2}}{a^{6/2}} = \frac{p^4}{a^3} \)
2. \( 2y^2 \sqrt{\frac{4x^2}{y^6}} = 2y^2 \frac{\sqrt{4x^2}}{\sqrt{y^6}} = 2y^2 \frac{2|x|}{y^{6/2}} = 2y^2 \frac{2x}{y^3} = \frac{4xy^2}{y^3} = \frac{4x}{y} \) (так как \( x > 0 \) и \( y > 0 \))
3. \( \sqrt[6]{\frac{16a^{12}}{b^{10}}} = \frac{\sqrt[6]{16a^{12}}}{\sqrt[6]{b^{10}}} = \frac{\sqrt[6]{16} a^{12/6}}{b^{10/6}} = \frac{\sqrt[6]{16} a^2}{b^{5/3}} \)
4. \( 3c^2 \sqrt[4]{\frac{c^6}{9d^2}} = 3c^2 \frac{\sqrt[4]{c^6}}{\sqrt[4]{9d^2}} = 3c^2 \frac{c^{6/4}}{\sqrt{\sqrt{9d^2}}} = 3c^2 \frac{c^{3/2}}{\sqrt{3|d|}} = \frac{3c^{7/2}}{\sqrt{3d}} \) (так как \( c < 0 \) и \( d > 0 \), \( \sqrt[4]{c^6} = \sqrt{|c^3|} = \sqrt{c^3} \) , \( \sqrt[4]{9d^2} = \sqrt{3d} \) )