Solve 5/sqrt{2+sqrt{3}}

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

Evaluate

$$5\left(\sqrt{3}-\sqrt{2}\right)\approx 1.589186226$$

Solution Steps

Rationalize the denominator of \frac{5}{\sqrt{2}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{3}.

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

Consider \left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\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}.

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

Square \sqrt{2}. Square \sqrt{3}.

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

Subtract 3 from 2 to get -1.

$$\frac{5\left(\sqrt{2}-\sqrt{3}\right)}{-1}$$

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply -5 by \sqrt{2}-\sqrt{3}.

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