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

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

Evaluate

$$\frac{5\sqrt{3}+7}{4}\approx 3.915063509$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4 from 12 to get 8.

$$\frac{\left(4+\sqrt{3}\right)\left(2\sqrt{3}+2\right)}{8}$$

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

$$\frac{8\sqrt{3}+8+2\left(\sqrt{3}\right)^{2}+2\sqrt{3}}{8}$$

The square of \sqrt{3} is 3.

$$\frac{8\sqrt{3}+8+2\times 3+2\sqrt{3}}{8}$$

Multiply 2 and 3 to get 6.

$$\frac{8\sqrt{3}+8+6+2\sqrt{3}}{8}$$

Add 8 and 6 to get 14.

$$\frac{8\sqrt{3}+14+2\sqrt{3}}{8}$$

Combine 8\sqrt{3} and 2\sqrt{3} to get 10\sqrt{3}.

$$\frac{10\sqrt{3}+14}{8}$$