Solve 2/sqrt{7+sqrt{3}}

$$\frac{ 2 }{ \sqrt{ 7 } + \sqrt{ 3 } }$$

Evaluate

$$\frac{\sqrt{7}-\sqrt{3}}{2}\approx 0.456850252$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{7}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{7}-\sqrt{3}.

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

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}.

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

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

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

Subtract 3 from 7 to get 4.

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

Divide 2\left(\sqrt{7}-\sqrt{3}\right) by 4 to get \frac{1}{2}\left(\sqrt{7}-\sqrt{3}\right).

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

Use the distributive property to multiply \frac{1}{2} by \sqrt{7}-\sqrt{3}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{1}{2}\sqrt{7}-\frac{1}{2}\sqrt{3}$$