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

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

Evaluate

$$1$$

Short Solution Steps

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

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

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

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

Square 3. Square \sqrt{3}.

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{9-3}+\frac{2-\sqrt{3}}{3-\sqrt{3}}$$

Subtract 3 from 9 to get 6.

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}+\frac{2-\sqrt{3}}{3-\sqrt{3}}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}+\frac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3-\sqrt{3}\right)\left(3+\sqrt{3}\right)}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}+\frac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{3^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 3. Square \sqrt{3}.

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}+\frac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{9-3}$$

Subtract 3 from 9 to get 6.

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6}+\frac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{6}$$

Since \frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)}{6} and \frac{\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{6}$$

Do the multiplications in \left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right).

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

Do the calculations in 6-2\sqrt{3}+3\sqrt{3}-3+6+2\sqrt{3}-3\sqrt{3}-3.

$$\frac{6}{6}$$

Divide 6 by 6 to get 1.

$$1$$

Factor

$$1$$