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

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

Evaluate

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

Solution Steps

Divide \sqrt{7}-\sqrt{3} by \sqrt{7}-\sqrt{3} to get 1.

$$\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}\times 1$$

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

$$\frac{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)}{\left(\sqrt{7}+\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)}\times 1$$

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

$$\frac{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{3}\right)^{2}}\times 1$$

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

$$\frac{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)}{7-3}\times 1$$

Subtract 3 from 7 to get 4.

$$\frac{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)}{4}\times 1$$

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

$$\frac{\left(\sqrt{7}-\sqrt{3}\right)^{2}}{4}\times 1$$

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

$$\frac{\left(\sqrt{7}\right)^{2}-2\sqrt{7}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{4}\times 1$$

The square of \sqrt{7} is 7.

$$\frac{7-2\sqrt{7}\sqrt{3}+\left(\sqrt{3}\right)^{2}}{4}\times 1$$

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

$$\frac{7-2\sqrt{21}+\left(\sqrt{3}\right)^{2}}{4}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{7-2\sqrt{21}+3}{4}\times 1$$

Add 7 and 3 to get 10.

$$\frac{10-2\sqrt{21}}{4}\times 1$$

Express \frac{10-2\sqrt{21}}{4}\times 1 as a single fraction.

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

Factor

$$\frac{5 - \sqrt{21}}{2} = 0.20871215252208009$$