Resolver =3/38+12sqrt{2}+5/38-12sqrt{2}

$$= \frac { 3 } { 38 + 12 \sqrt { 2 } } + \frac { 5 } { 38 - 12 \sqrt { 2 } }$$

Calcular

$$\frac{6\sqrt{2}+76}{289}\approx 0.292336614$$

Pasos de solución curta

Racionaliza o denominador de \frac{3}{38+12\sqrt{2}} mediante a multiplicación do numerador e o denominador por 38-12\sqrt{2}.

$$\frac{3\left(38-12\sqrt{2}\right)}{\left(38+12\sqrt{2}\right)\left(38-12\sqrt{2}\right)}+\frac{5}{38-12\sqrt{2}}$$

Considera \left(38+12\sqrt{2}\right)\left(38-12\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}.

$$\frac{3\left(38-12\sqrt{2}\right)}{38^{2}-\left(12\sqrt{2}\right)^{2}}+\frac{5}{38-12\sqrt{2}}$$

Calcula 38 á potencia de 2 e obtén 1444.

$$\frac{3\left(38-12\sqrt{2}\right)}{1444-\left(12\sqrt{2}\right)^{2}}+\frac{5}{38-12\sqrt{2}}$$

Expande \left(12\sqrt{2}\right)^{2}.

$$\frac{3\left(38-12\sqrt{2}\right)}{1444-12^{2}\left(\sqrt{2}\right)^{2}}+\frac{5}{38-12\sqrt{2}}$$

Calcula 12 á potencia de 2 e obtén 144.

$$\frac{3\left(38-12\sqrt{2}\right)}{1444-144\left(\sqrt{2}\right)^{2}}+\frac{5}{38-12\sqrt{2}}$$

O cadrado de \sqrt{2} é 2.

$$\frac{3\left(38-12\sqrt{2}\right)}{1444-144\times 2}+\frac{5}{38-12\sqrt{2}}$$

Multiplica 144 e 2 para obter 288.

$$\frac{3\left(38-12\sqrt{2}\right)}{1444-288}+\frac{5}{38-12\sqrt{2}}$$

Resta 288 de 1444 para obter 1156.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5}{38-12\sqrt{2}}$$

Racionaliza o denominador de \frac{5}{38-12\sqrt{2}} mediante a multiplicación do numerador e o denominador por 38+12\sqrt{2}.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{\left(38-12\sqrt{2}\right)\left(38+12\sqrt{2}\right)}$$

Considera \left(38-12\sqrt{2}\right)\left(38+12\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}.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{38^{2}-\left(-12\sqrt{2}\right)^{2}}$$

Calcula 38 á potencia de 2 e obtén 1444.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1444-\left(-12\sqrt{2}\right)^{2}}$$

Expande \left(-12\sqrt{2}\right)^{2}.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1444-\left(-12\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calcula -12 á potencia de 2 e obtén 144.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1444-144\left(\sqrt{2}\right)^{2}}$$

O cadrado de \sqrt{2} é 2.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1444-144\times 2}$$

Multiplica 144 e 2 para obter 288.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1444-288}$$

Resta 288 de 1444 para obter 1156.

$$\frac{3\left(38-12\sqrt{2}\right)}{1156}+\frac{5\left(38+12\sqrt{2}\right)}{1156}$$

Dado que \frac{3\left(38-12\sqrt{2}\right)}{1156} e \frac{5\left(38+12\sqrt{2}\right)}{1156} teñen o mesmo denominador, súmaos mediante a suma dos seus numeradores.

$$\frac{3\left(38-12\sqrt{2}\right)+5\left(38+12\sqrt{2}\right)}{1156}$$

Fai as multiplicacións en 3\left(38-12\sqrt{2}\right)+5\left(38+12\sqrt{2}\right).

$$\frac{114-36\sqrt{2}+190+60\sqrt{2}}{1156}$$

Fai os cálculos en 114-36\sqrt{2}+190+60\sqrt{2}.

$$\frac{304+24\sqrt{2}}{1156}$$

Factorizar

$$\frac{2 {(3 \sqrt{2} + 38)}}{289} = 0.29233661375169057$$