Resolver [(sqrt{2})^{2}*(2/sqrt{3})^2][(1/2)^2+4*1^2-2^2]

$$[ ( \sqrt { 2 } ) ^ { 2 } \times ( \frac { 2 } { \sqrt { 3 } } ) ^ { 2 } ] [ ( \frac { 1 } { 2 } ) ^ { 2 } + 4 \times 1 ^ { 2 } - 2 ^ { 2 } ]$$

Calcular

$$\frac{2}{3}\approx 0.666666667$$

Pasos da solución

O cadrado de \sqrt{2} é 2.

$$2\times \left(\frac{2}{\sqrt{3}}\right)^{2}\left(\left(\frac{1}{2}\right)^{2}+4\times 1^{2}-2^{2}\right)$$

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

$$2\times \left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{2}\left(\left(\frac{1}{2}\right)^{2}+4\times 1^{2}-2^{2}\right)$$

O cadrado de \sqrt{3} é 3.

$$2\times \left(\frac{2\sqrt{3}}{3}\right)^{2}\left(\left(\frac{1}{2}\right)^{2}+4\times 1^{2}-2^{2}\right)$$

Para elevar \frac{2\sqrt{3}}{3} a unha potencia, eleva o numerador e o denominador á potencia e despois divide.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\left(\frac{1}{2}\right)^{2}+4\times 1^{2}-2^{2}\right)$$

Calcula \frac{1}{2} á potencia de 2 e obtén \frac{1}{4}.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{1}{4}+4\times 1^{2}-2^{2}\right)$$

Calcula 1 á potencia de 2 e obtén 1.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{1}{4}+4\times 1-2^{2}\right)$$

Multiplica 4 e 1 para obter 4.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{1}{4}+4-2^{2}\right)$$

Suma \frac{1}{4} e 4 para obter \frac{17}{4}.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{17}{4}-2^{2}\right)$$

Calcula 2 á potencia de 2 e obtén 4.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\left(\frac{17}{4}-4\right)$$

Resta 4 de \frac{17}{4} para obter \frac{1}{4}.

$$2\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}\times \frac{1}{4}$$

Multiplica 2 e \frac{1}{4} para obter \frac{1}{2}.

$$\frac{1}{2}\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}$$

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

$$\frac{1}{2}\times \frac{2^{2}\left(\sqrt{3}\right)^{2}}{3^{2}}$$

Calcula 2 á potencia de 2 e obtén 4.

$$\frac{1}{2}\times \frac{4\left(\sqrt{3}\right)^{2}}{3^{2}}$$

O cadrado de \sqrt{3} é 3.

$$\frac{1}{2}\times \frac{4\times 3}{3^{2}}$$

Multiplica 4 e 3 para obter 12.

$$\frac{1}{2}\times \frac{12}{3^{2}}$$

Calcula 3 á potencia de 2 e obtén 9.

$$\frac{1}{2}\times \frac{12}{9}$$

Reduce a fracción \frac{12}{9} a termos máis baixos extraendo e cancelando 3.

$$\frac{1}{2}\times \frac{4}{3}$$

Multiplica \frac{1}{2} e \frac{4}{3} para obter \frac{2}{3}.

$$\frac{2}{3}$$

Factorizar

$$\frac{2}{3} = 0.6666666666666666$$