Solve frac{sqrt{3}}{4-2sqrt{5}}

$$\frac { \sqrt { 3 } } { 4 - 2 \sqrt { 5 } }$$

Evaluate

$$-\frac{\sqrt{3}\left(\sqrt{5}+2\right)}{2}\approx -3.668542481$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}}{4-2\sqrt{5}} by multiplying numerator and denominator by 4+2\sqrt{5}.

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

Consider \left(4-2\sqrt{5}\right)\left(4+2\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{\sqrt{3}\left(4+2\sqrt{5}\right)}{16-\left(-2\sqrt{5}\right)^{2}}$$

Expand \left(-2\sqrt{5}\right)^{2}.

$$\frac{\sqrt{3}\left(4+2\sqrt{5}\right)}{16-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\sqrt{3}\left(4+2\sqrt{5}\right)}{16-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\left(4+2\sqrt{5}\right)}{16-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{\sqrt{3}\left(4+2\sqrt{5}\right)}{16-20}$$

Subtract 20 from 16 to get -4.

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

Use the distributive property to multiply \sqrt{3} by 4+2\sqrt{5}.

$$\frac{4\sqrt{3}+2\sqrt{3}\sqrt{5}}{-4}$$

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

$$\frac{4\sqrt{3}+2\sqrt{15}}{-4}$$