Resolver 5/2sqrt{7-2sqrt{5}}

$$\frac { 5 } { 2 \sqrt { 7 } - 2 \sqrt { 5 } }$$

Calcular

$$\frac{5\left(\sqrt{5}+\sqrt{7}\right)}{4}\approx 6.102274111$$

Pasos da solución

Racionaliza o denominador de \frac{5}{2\sqrt{7}-2\sqrt{5}} mediante a multiplicación do numerador e o denominador por 2\sqrt{7}+2\sqrt{5}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{\left(2\sqrt{7}-2\sqrt{5}\right)\left(2\sqrt{7}+2\sqrt{5}\right)}$$

Considera \left(2\sqrt{7}-2\sqrt{5}\right)\left(2\sqrt{7}+2\sqrt{5}\right). A multiplicación pódese transformar na diferencia de cadrados mediante a regra: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{\left(2\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Expande \left(2\sqrt{7}\right)^{2}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{2^{2}\left(\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

Calcula 2 á potencia de 2 e obtén 4.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{4\left(\sqrt{7}\right)^{2}-\left(-2\sqrt{5}\right)^{2}}$$

O cadrado de \sqrt{7} é 7.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{4\times 7-\left(-2\sqrt{5}\right)^{2}}$$

Multiplica 4 e 7 para obter 28.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-\left(-2\sqrt{5}\right)^{2}}$$

Expande \left(-2\sqrt{5}\right)^{2}.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calcula -2 á potencia de 2 e obtén 4.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-4\left(\sqrt{5}\right)^{2}}$$

O cadrado de \sqrt{5} é 5.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-4\times 5}$$

Multiplica 4 e 5 para obter 20.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{28-20}$$

Resta 20 de 28 para obter 8.

$$\frac{5\left(2\sqrt{7}+2\sqrt{5}\right)}{8}$$

Usa a propiedade distributiva para multiplicar 5 por 2\sqrt{7}+2\sqrt{5}.

$$\frac{10\sqrt{7}+10\sqrt{5}}{8}$$