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

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

Evaluate

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

Short Solution Steps

Rationalize the denominator of \frac{3}{38+12\sqrt{2}} by multiplying numerator and denominator by 38-12\sqrt{2}.

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

Consider \left(38+12\sqrt{2}\right)\left(38-12\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 38 to the power of 2 and get 1444.

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

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

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

Calculate 12 to the power of 2 and get 144.

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

The square of \sqrt{2} is 2.

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

Multiply 144 and 2 to get 288.

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

Subtract 288 from 1444 to get 1156.

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

Rationalize the denominator of \frac{5}{38-12\sqrt{2}} by multiplying numerator and denominator by 38+12\sqrt{2}.

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

Consider \left(38-12\sqrt{2}\right)\left(38+12\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 38 to the power of 2 and get 1444.

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

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

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

Calculate -12 to the power of 2 and get 144.

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

The square of \sqrt{2} is 2.

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

Multiply 144 and 2 to get 288.

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

Subtract 288 from 1444 to get 1156.

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

Since \frac{3\left(38-12\sqrt{2}\right)}{1156} and \frac{5\left(38+12\sqrt{2}\right)}{1156} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(38-12\sqrt{2}\right)+5\left(38+12\sqrt{2}\right).

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

Do the calculations in 114-36\sqrt{2}+190+60\sqrt{2}.

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

Factor

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