Resolver frac{2sqrt{3}-sqrt{2}}{2sqrt{3}+sqrt{2}}

$$\frac { 2 \sqrt { 3 } - \sqrt { 2 } } { 2 \sqrt { 3 } + \sqrt { 2 } }$$

Calcular

$$\frac{7-2\sqrt{6}}{5}\approx 0.420204103$$

Pasos da solución

Racionaliza o denominador de \frac{2\sqrt{3}-\sqrt{2}}{2\sqrt{3}+\sqrt{2}} mediante a multiplicación do numerador e o denominador por 2\sqrt{3}-\sqrt{2}.

$$\frac{\left(2\sqrt{3}-\sqrt{2}\right)\left(2\sqrt{3}-\sqrt{2}\right)}{\left(2\sqrt{3}+\sqrt{2}\right)\left(2\sqrt{3}-\sqrt{2}\right)}$$

Considera \left(2\sqrt{3}+\sqrt{2}\right)\left(2\sqrt{3}-\sqrt{2}\right). A multiplicación pódese transformar na diferencia de cadrados mediante a regra: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{3}-\sqrt{2}\right)\left(2\sqrt{3}-\sqrt{2}\right)}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Multiplica 2\sqrt{3}-\sqrt{2} e 2\sqrt{3}-\sqrt{2} para obter \left(2\sqrt{3}-\sqrt{2}\right)^{2}.

$$\frac{\left(2\sqrt{3}-\sqrt{2}\right)^{2}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Usar teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(2\sqrt{3}-\sqrt{2}\right)^{2}.

$$\frac{4\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

O cadrado de \sqrt{3} é 3.

$$\frac{4\times 3-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Multiplica 4 e 3 para obter 12.

$$\frac{12-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Para multiplicar \sqrt{3} e \sqrt{2}, multiplica os números baixo a raíz cadrada.

$$\frac{12-4\sqrt{6}+\left(\sqrt{2}\right)^{2}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

O cadrado de \sqrt{2} é 2.

$$\frac{12-4\sqrt{6}+2}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Suma 12 e 2 para obter 14.

$$\frac{14-4\sqrt{6}}{\left(2\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Expande \left(2\sqrt{3}\right)^{2}.

$$\frac{14-4\sqrt{6}}{2^{2}\left(\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Calcula 2 á potencia de 2 e obtén 4.

$$\frac{14-4\sqrt{6}}{4\left(\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

O cadrado de \sqrt{3} é 3.

$$\frac{14-4\sqrt{6}}{4\times 3-\left(\sqrt{2}\right)^{2}}$$

Multiplica 4 e 3 para obter 12.

$$\frac{14-4\sqrt{6}}{12-\left(\sqrt{2}\right)^{2}}$$

O cadrado de \sqrt{2} é 2.

$$\frac{14-4\sqrt{6}}{12-2}$$

Resta 2 de 12 para obter 10.

$$\frac{14-4\sqrt{6}}{10}$$

Factorizar

$$\frac{7 - 2 \sqrt{6}}{5} = 0.4202041028867288$$