Solve frac{sqrt{75}-sqrt{18}}{sqrt{12}}

$$\frac { \sqrt { 75 } - \sqrt { 18 } } { \sqrt { 12 } }$$

Evaluate

$$\frac{5-\sqrt{6}}{2}\approx 1.275255129$$

Solution Steps

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{5\sqrt{3}-\sqrt{18}}{\sqrt{12}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{5\sqrt{3}-3\sqrt{2}}{\sqrt{12}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

$$\frac{\left(5\sqrt{3}-3\sqrt{2}\right)\sqrt{3}}{6}$$

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

$$\frac{5\left(\sqrt{3}\right)^{2}-3\sqrt{2}\sqrt{3}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{5\times 3-3\sqrt{2}\sqrt{3}}{6}$$

Multiply 5 and 3 to get 15.

$$\frac{15-3\sqrt{2}\sqrt{3}}{6}$$

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

$$\frac{15-3\sqrt{6}}{6}$$