Mathematic Problems

277. példa

\frac{{(\sqrt{a} - \sqrt{b})}^{3} + 2 \sqrt{a^{3}} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} + \frac{3 \sqrt{a b} - b}{a - b} a > 0; b > 0

\sqrt{a} = x; \sqrt{b} = y

\frac{{(x - y)}^{3} + 2 x^{3} + y^{3}}{x^{3} + y^{3}} + \frac{3 x y - 3 y^{2}}{x^{2} - y^{2}}

= \frac{x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3} + 2 x^{3} + y^{3}}{(x + y) (x^{2} - x y + y^{2})} + \frac{3 x y - 3 y^{2}}{(x - y) (x + y)}

= \frac{3 x (x^{2} - x y + y^{2})}{(x + y) (x^{2} - x y + y^{2})} + \frac{3 y (x - y)}{(x - y) (x + y)}

= \frac{3 x + 3 y}{(x + y)}

= \frac{3 (x + y)}{(x + y)}

= 3

\frac{{(\sqrt{a} - \sqrt{b})}^{3} + 2 \sqrt{a^{3}} + b \sqrt{b}}{a \sqrt{a} + b \sqrt{b}} + \frac{3 \sqrt{a b} - b}{a - b} = 3

Matematika - Képlettár