Solve frac{4sqrt{3}+4}{1+sqrt{3}}

$$\frac { 4 \sqrt { 3 } + 4 } { 1 + \sqrt { 3 } }$$

Evaluate

$$4$$

Solution Steps

Rationalize the denominator of \frac{4\sqrt{3}+4}{1+\sqrt{3}} by multiplying numerator and denominator by 1-\sqrt{3}.

$$\frac{\left(4\sqrt{3}+4\right)\left(1-\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}$$

Consider \left(1+\sqrt{3}\right)\left(1-\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}.

$$\frac{\left(4\sqrt{3}+4\right)\left(1-\sqrt{3}\right)}{1^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 1. Square \sqrt{3}.

$$\frac{\left(4\sqrt{3}+4\right)\left(1-\sqrt{3}\right)}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{\left(4\sqrt{3}+4\right)\left(1-\sqrt{3}\right)}{-2}$$

Apply the distributive property by multiplying each term of 4\sqrt{3}+4 by each term of 1-\sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply -4 and 3 to get -12.

$$\frac{4\sqrt{3}-12+4-4\sqrt{3}}{-2}$$

Add -12 and 4 to get -8.

$$\frac{4\sqrt{3}-8-4\sqrt{3}}{-2}$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$\frac{-8}{-2}$$

Divide -8 by -2 to get 4.

$$4$$

Factor

$$2^{2}$$