Solve frac{sqrt{5}-sqrt{7}}{sqrt{5}+sqrt{7}}

$$\frac{ \sqrt{ 5 } - \sqrt{ 7 } }{ \sqrt{ 5 } + \sqrt{ 7 } }$$

Evaluate

$$\sqrt{35}-6\approx -0.083920217$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{7}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}{\left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}$$

Consider \left(\sqrt{5}+\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{7}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}{5-7}$$

Subtract 7 from 5 to get -2.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)\left(\sqrt{5}-\sqrt{7}\right)}{-2}$$

Multiply \sqrt{5}-\sqrt{7} and \sqrt{5}-\sqrt{7} to get \left(\sqrt{5}-\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{5}-\sqrt{7}\right)^{2}}{-2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{5}\right)^{2}-2\sqrt{5}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{-2}$$

The square of \sqrt{5} is 5.

$$\frac{5-2\sqrt{5}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{-2}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{5-2\sqrt{35}+\left(\sqrt{7}\right)^{2}}{-2}$$

The square of \sqrt{7} is 7.

$$\frac{5-2\sqrt{35}+7}{-2}$$

Add 5 and 7 to get 12.

$$\frac{12-2\sqrt{35}}{-2}$$

Divide each term of 12-2\sqrt{35} by -2 to get -6+\sqrt{35}.

$$-6+\sqrt{35}$$