Solve frac{sqrt{7}}{sqrt{14}+sqrt{7}}

$$\frac { \sqrt { 7 } } { \sqrt { 14 } + \sqrt { 7 } }$$

Evaluate

$$\sqrt{2}-1\approx 0.414213562$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{14}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{14}-\sqrt{7}.

$$\frac{\sqrt{7}\left(\sqrt{14}-\sqrt{7}\right)}{\left(\sqrt{14}+\sqrt{7}\right)\left(\sqrt{14}-\sqrt{7}\right)}$$

Consider \left(\sqrt{14}+\sqrt{7}\right)\left(\sqrt{14}-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{7}\left(\sqrt{14}-\sqrt{7}\right)}{\left(\sqrt{14}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{14}. Square \sqrt{7}.

$$\frac{\sqrt{7}\left(\sqrt{14}-\sqrt{7}\right)}{14-7}$$

Subtract 7 from 14 to get 7.

$$\frac{\sqrt{7}\left(\sqrt{14}-\sqrt{7}\right)}{7}$$

Use the distributive property to multiply \sqrt{7} by \sqrt{14}-\sqrt{7}.

$$\frac{\sqrt{7}\sqrt{14}-\left(\sqrt{7}\right)^{2}}{7}$$

Factor 14=7\times 2. Rewrite the square root of the product \sqrt{7\times 2} as the product of square roots \sqrt{7}\sqrt{2}.

$$\frac{\sqrt{7}\sqrt{7}\sqrt{2}-\left(\sqrt{7}\right)^{2}}{7}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\frac{7\sqrt{2}-\left(\sqrt{7}\right)^{2}}{7}$$

The square of \sqrt{7} is 7.

$$\frac{7\sqrt{2}-7}{7}$$

Divide each term of 7\sqrt{2}-7 by 7 to get \sqrt{2}-1.

$$\sqrt{2}-1$$