Solve frac{sqrt{50}-2sqrt{3}}{sqrt{2}}

$$\frac { \sqrt { 50 } - 2 \sqrt { 3 } } { \sqrt { 2 } }$$

Evaluate

$$5-\sqrt{6}\approx 2.550510257$$

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}.

$$\frac{5\sqrt{2}-2\sqrt{3}}{\sqrt{2}}$$

Rationalize the denominator of \frac{5\sqrt{2}-2\sqrt{3}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply 5\sqrt{2}-2\sqrt{3} by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{5\times 2-2\sqrt{3}\sqrt{2}}{2}$$

Multiply 5 and 2 to get 10.

$$\frac{10-2\sqrt{3}\sqrt{2}}{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{10-2\sqrt{6}}{2}$$

Divide each term of 10-2\sqrt{6} by 2 to get 5-\sqrt{6}.

$$5-\sqrt{6}$$