Solve frac{sqrt{3}-12}{sqrt{20}*sqrt{3}}

$$\frac { \sqrt { 3 } - 12 } { \sqrt { 20 } \cdot \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{5}-4\sqrt{15}}{10}\approx -1.325586541$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{3}-12}{2\sqrt{15}}$$

Rationalize the denominator of \frac{\sqrt{3}-12}{2\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\left(\sqrt{3}-12\right)\sqrt{15}}{2\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{\left(\sqrt{3}-12\right)\sqrt{15}}{2\times 15}$$

Multiply 2 and 15 to get 30.

$$\frac{\left(\sqrt{3}-12\right)\sqrt{15}}{30}$$

Use the distributive property to multiply \sqrt{3}-12 by \sqrt{15}.

$$\frac{\sqrt{3}\sqrt{15}-12\sqrt{15}}{30}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{\sqrt{3}\sqrt{3}\sqrt{5}-12\sqrt{15}}{30}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{5}-12\sqrt{15}}{30}$$