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

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

Evaluate

$$\sqrt{15}-4\approx -0.127016654$$

Solution Steps

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

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

Consider \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

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

Apply the distributive property by multiplying each term of \sqrt{3}-\sqrt{5} by each term of \sqrt{5}-\sqrt{3}.

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

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

$$\frac{\sqrt{15}-\left(\sqrt{3}\right)^{2}-\left(\sqrt{5}\right)^{2}+\sqrt{3}\sqrt{5}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{15}-3-\left(\sqrt{5}\right)^{2}+\sqrt{3}\sqrt{5}}{2}$$

The square of \sqrt{5} is 5.

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

Subtract 5 from -3 to get -8.

$$\frac{\sqrt{15}-8+\sqrt{3}\sqrt{5}}{2}$$

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

$$\frac{\sqrt{15}-8+\sqrt{15}}{2}$$

Combine \sqrt{15} and \sqrt{15} to get 2\sqrt{15}.

$$\frac{2\sqrt{15}-8}{2}$$

Divide each term of 2\sqrt{15}-8 by 2 to get \sqrt{15}-4.

$$\sqrt{15}-4$$