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

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

Evaluate

$$6-2\sqrt{3}\approx 2.535898385$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{3}.

$$4\sqrt{3}\times \frac{1-\sqrt{3}}{1-3}$$

Subtract 3 from 1 to get -2.

$$4\sqrt{3}\times \frac{1-\sqrt{3}}{-2}$$

Multiply both numerator and denominator by -1.

$$4\sqrt{3}\times \frac{-1+\sqrt{3}}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

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

Use the distributive property to multiply 2 by -1+\sqrt{3}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

$$-2\sqrt{3}+6$$