Solve frac{2-sqrt{7}}{3+sqrt{7}}=

$$\frac { 2 - \sqrt { 7 } } { 3 + \sqrt { 7 } } =$$

Evaluate

$$\frac{13-5\sqrt{7}}{2}\approx -0.114378278$$

Solution Steps

Rationalize the denominator of \frac{2-\sqrt{7}}{3+\sqrt{7}} by multiplying numerator and denominator by 3-\sqrt{7}.

$$\frac{\left(2-\sqrt{7}\right)\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}$$

Consider \left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2-\sqrt{7}\right)\left(3-\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 3. Square \sqrt{7}.

$$\frac{\left(2-\sqrt{7}\right)\left(3-\sqrt{7}\right)}{9-7}$$

Subtract 7 from 9 to get 2.

$$\frac{\left(2-\sqrt{7}\right)\left(3-\sqrt{7}\right)}{2}$$

Apply the distributive property by multiplying each term of 2-\sqrt{7} by each term of 3-\sqrt{7}.

$$\frac{6-2\sqrt{7}-3\sqrt{7}+\left(\sqrt{7}\right)^{2}}{2}$$

Combine -2\sqrt{7} and -3\sqrt{7} to get -5\sqrt{7}.

$$\frac{6-5\sqrt{7}+\left(\sqrt{7}\right)^{2}}{2}$$

The square of \sqrt{7} is 7.

$$\frac{6-5\sqrt{7}+7}{2}$$

Add 6 and 7 to get 13.

$$\frac{13-5\sqrt{7}}{2}$$