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

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

Evaluate

$$\frac{4\sqrt{15}}{3}-2\sqrt{5}\approx 0.69184184$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

$$\frac{2\sqrt{5}\left(3-2\sqrt{3}\right)}{9-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{2\sqrt{5}\left(3-2\sqrt{3}\right)}{9-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{5}\left(3-2\sqrt{3}\right)}{9-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{5}\left(3-2\sqrt{3}\right)}{9-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{2\sqrt{5}\left(3-2\sqrt{3}\right)}{9-12}$$

Subtract 12 from 9 to get -3.

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

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

$$\frac{6\sqrt{5}-4\sqrt{3}\sqrt{5}}{-3}$$

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

$$\frac{6\sqrt{5}-4\sqrt{15}}{-3}$$