Solve frac{1+sqrt{3}}{2+sqrt{3}}

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

Evaluate

$$\sqrt{3}-1\approx 0.732050808$$

Solution Steps

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

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

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

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of 1+\sqrt{3} by each term of 2-\sqrt{3}.

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

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

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 2 to get -1.

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