Solve frac{sqrt{3}-sqrt{7}}{sqrt{3}+sqrt{7}}

$$\frac { \sqrt { 3 } - \sqrt { 7 } } { \sqrt { 3 } + \sqrt { 7 } }$$

Evaluate

$$\frac{\sqrt{21}-5}{2}\approx -0.208712153$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}-\sqrt{7}}{\sqrt{3}+\sqrt{7}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{7}.

$$\frac{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}{\left(\sqrt{3}+\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}$$

Consider \left(\sqrt{3}+\sqrt{7}\right)\left(\sqrt{3}-\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}.

$$\frac{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}{\left(\sqrt{3}\right)^{2}-\left(\sqrt{7}\right)^{2}}$$

Square \sqrt{3}. Square \sqrt{7}.

$$\frac{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}{3-7}$$

Subtract 7 from 3 to get -4.

$$\frac{\left(\sqrt{3}-\sqrt{7}\right)\left(\sqrt{3}-\sqrt{7}\right)}{-4}$$

Multiply \sqrt{3}-\sqrt{7} and \sqrt{3}-\sqrt{7} to get \left(\sqrt{3}-\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{3}-\sqrt{7}\right)^{2}}{-4}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{3}\right)^{2}-2\sqrt{3}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{-4}$$

The square of \sqrt{3} is 3.

$$\frac{3-2\sqrt{3}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{-4}$$

To multiply \sqrt{3} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{3-2\sqrt{21}+\left(\sqrt{7}\right)^{2}}{-4}$$

The square of \sqrt{7} is 7.

$$\frac{3-2\sqrt{21}+7}{-4}$$

Add 3 and 7 to get 10.

$$\frac{10-2\sqrt{21}}{-4}$$