Solve frac{(sqrt{3}-1)*(sqrt{3}+1)}{(sqrt{3}+1)*(sqrt{3}+1)}+sqrt{3}

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

Evaluate

$$2$$

Solution Steps

Multiply \sqrt{3}+1 and \sqrt{3}+1 to get \left(\sqrt{3}+1\right)^{2}.

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

Cancel out \sqrt{3}+1 in both numerator and denominator.

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

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

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

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}.

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

Square \sqrt{3}. Square 1.

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

Subtract 1 from 3 to get 2.

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

Multiply \sqrt{3}-1 and \sqrt{3}-1 to get \left(\sqrt{3}-1\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-1\right)^{2}.

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

The square of \sqrt{3} is 3.

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

Add 3 and 1 to get 4.

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

Divide each term of 4-2\sqrt{3} by 2 to get 2-\sqrt{3}.

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

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

$$2$$

Factor

$$2$$