Solve frac{3+sqrt{14}}{8+sqrt{14}}

$$\frac { 3 + \sqrt { 14 } } { 8 + \sqrt { 14 } }$$

Evaluate

$$\frac{\sqrt{14}+2}{10}\approx 0.574165739$$

Solution Steps

Rationalize the denominator of \frac{3+\sqrt{14}}{8+\sqrt{14}} by multiplying numerator and denominator by 8-\sqrt{14}.

$$\frac{\left(3+\sqrt{14}\right)\left(8-\sqrt{14}\right)}{\left(8+\sqrt{14}\right)\left(8-\sqrt{14}\right)}$$

Consider \left(8+\sqrt{14}\right)\left(8-\sqrt{14}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3+\sqrt{14}\right)\left(8-\sqrt{14}\right)}{8^{2}-\left(\sqrt{14}\right)^{2}}$$

Square 8. Square \sqrt{14}.

$$\frac{\left(3+\sqrt{14}\right)\left(8-\sqrt{14}\right)}{64-14}$$

Subtract 14 from 64 to get 50.

$$\frac{\left(3+\sqrt{14}\right)\left(8-\sqrt{14}\right)}{50}$$

Apply the distributive property by multiplying each term of 3+\sqrt{14} by each term of 8-\sqrt{14}.

$$\frac{24-3\sqrt{14}+8\sqrt{14}-\left(\sqrt{14}\right)^{2}}{50}$$

Combine -3\sqrt{14} and 8\sqrt{14} to get 5\sqrt{14}.

$$\frac{24+5\sqrt{14}-\left(\sqrt{14}\right)^{2}}{50}$$

The square of \sqrt{14} is 14.

$$\frac{24+5\sqrt{14}-14}{50}$$

Subtract 14 from 24 to get 10.

$$\frac{10+5\sqrt{14}}{50}$$