Solve frac{3+2sqrt{3}}{2sqrt{3}+4}

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

Evaluate

$$\frac{\sqrt{3}}{2}\approx 0.866025404$$

Solution Steps

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

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

Consider \left(2\sqrt{3}+4\right)\left(2\sqrt{3}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$\frac{\left(3+2\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{12-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(3+2\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{12-16}$$

Subtract 16 from 12 to get -4.

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

Apply the distributive property by multiplying each term of 3+2\sqrt{3} by each term of 2\sqrt{3}-4.

$$\frac{6\sqrt{3}-12+4\left(\sqrt{3}\right)^{2}-8\sqrt{3}}{-4}$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{3}-12+4\times 3-8\sqrt{3}}{-4}$$

Multiply 4 and 3 to get 12.

$$\frac{6\sqrt{3}-12+12-8\sqrt{3}}{-4}$$

Add -12 and 12 to get 0.

$$\frac{6\sqrt{3}-8\sqrt{3}}{-4}$$

Combine 6\sqrt{3} and -8\sqrt{3} to get -2\sqrt{3}.

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

Divide -2\sqrt{3} by -4 to get \frac{1}{2}\sqrt{3}.

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