Solve frac{sqrt{5}-sqrt{3}-sqrt{5}-sqrt{3}}{sqrt{3}+sqrt{3}*sqrt{5}-sqrt{3}}

$$\frac { \sqrt { 5 } - \sqrt { 3 } - \sqrt { 5 } - \sqrt { 3 } } { \sqrt { 3 } + \sqrt { 3 } \cdot \sqrt { 5 } - \sqrt { 3 } }$$

Evaluate

$$-\frac{2\sqrt{5}}{5}\approx -0.894427191$$

Solution Steps

Combine \sqrt{5} and -\sqrt{5} to get 0.

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

Combine -\sqrt{3} and -\sqrt{3} to get -2\sqrt{3}.

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

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

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

Combine \sqrt{3} and -\sqrt{3} to get 0.

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

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

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

The square of \sqrt{15} is 15.

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

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}.

$$\frac{-2\sqrt{3}\sqrt{3}\sqrt{5}}{15}$$

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

$$\frac{-2\times 3\sqrt{5}}{15}$$

Multiply -2 and 3 to get -6.

$$\frac{-6\sqrt{5}}{15}$$

Divide -6\sqrt{5} by 15 to get -\frac{2}{5}\sqrt{5}.

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