Solve 3/1-sqrt{7}

$$\frac { 3 } { 1 - \sqrt { 7 } }$$

Evaluate

$$\frac{-\sqrt{7}-1}{2}\approx -1.822875656$$

Solution Steps

Rationalize the denominator of \frac{3}{1-\sqrt{7}} by multiplying numerator and denominator by 1+\sqrt{7}.

$$\frac{3\left(1+\sqrt{7}\right)}{\left(1-\sqrt{7}\right)\left(1+\sqrt{7}\right)}$$

Consider \left(1-\sqrt{7}\right)\left(1+\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}.

$$\frac{3\left(1+\sqrt{7}\right)}{1^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 1. Square \sqrt{7}.

$$\frac{3\left(1+\sqrt{7}\right)}{1-7}$$

Subtract 7 from 1 to get -6.

$$\frac{3\left(1+\sqrt{7}\right)}{-6}$$

Divide 3\left(1+\sqrt{7}\right) by -6 to get -\frac{1}{2}\left(1+\sqrt{7}\right).

$$-\frac{1}{2}\left(1+\sqrt{7}\right)$$

Use the distributive property to multiply -\frac{1}{2} by 1+\sqrt{7}.

$$-\frac{1}{2}-\frac{1}{2}\sqrt{7}$$