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

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

Evaluate

$$\sqrt{3}\left(\sqrt{5}-2\right)\approx 0.408881731$$

Solution Steps

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

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

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

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

Square \sqrt{5}. Square 2.

$$\frac{\sqrt{3}\left(\sqrt{5}-2\right)}{5-4}$$

Subtract 4 from 5 to get 1.

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

Anything divided by one gives itself.

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

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

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

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\sqrt{15}-2\sqrt{3}$$