Solve frac{sqrt{50}-sqrt{14}}{sqrt{2}}

$$\frac{ \sqrt{ 50 } - \sqrt{ 14 } }{ \sqrt{ 2 } }$$

Evaluate

$$5-\sqrt{7}\approx 2.354248689$$

Solution Steps

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{5\sqrt{2}-\sqrt{14}}{\sqrt{2}}$$

Rationalize the denominator of \frac{5\sqrt{2}-\sqrt{14}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{\left(5\sqrt{2}-\sqrt{14}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 5\sqrt{2}-\sqrt{14} by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{5\times 2-\sqrt{14}\sqrt{2}}{2}$$

Multiply 5 and 2 to get 10.

$$\frac{10-\sqrt{14}\sqrt{2}}{2}$$

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

$$\frac{10-\sqrt{2}\sqrt{7}\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{10-2\sqrt{7}}{2}$$

Divide each term of 10-2\sqrt{7} by 2 to get 5-\sqrt{7}.

$$5-\sqrt{7}$$