Solve 2/sqrt{3-1}+sqrt{3}left(1+sqrt{3}right)-2sqrt{3}

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

Evaluate

$$4$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{3}-1} by multiplying numerator and denominator by \sqrt{3}+1.

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

Consider \left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Cancel out 2 and 2.

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

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

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

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

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

The square of \sqrt{3} is 3.

$$-\sqrt{3}+1+\sqrt{3}+3$$

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

$$1+3$$

Add 1 and 3 to get 4.

$$4$$

Factor

$$2^{2}$$