Solve frac{sqrt{14}+sqrt{7}}{sqrt{7}}-2sqrt{2}

$$\frac { \sqrt { 14 } + \sqrt { 7 } } { \sqrt { 7 } } - 2 \sqrt { 2 }$$

Evaluate

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

Short Solution Steps

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

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

The square of \sqrt{7} is 7.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -2\sqrt{2} times \frac{7}{7}.

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

Since \frac{\left(\sqrt{14}+\sqrt{7}\right)\sqrt{7}}{7} and \frac{7\left(-2\right)\sqrt{2}}{7} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(\sqrt{14}+\sqrt{7}\right)\sqrt{7}+7\left(-2\right)\sqrt{2}.

$$\frac{7\sqrt{2}+7-14\sqrt{2}}{7}$$

Do the calculations in 7\sqrt{2}+7-14\sqrt{2}.

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

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

$$-\sqrt{2}+1$$