Solve 13/1-sqrt{7}

$$\frac{ 13 }{ 1- \sqrt{ 7 } }$$

Evaluate

$$\frac{-13\sqrt{7}-13}{6}\approx -7.899127841$$

Solution Steps

Rationalize the denominator of \frac{13}{1-\sqrt{7}} by multiplying numerator and denominator by 1+\sqrt{7}.

$$\frac{13\left(1+\sqrt{7}\right)}{\left(1-\sqrt{7}\right)\left(1+\sqrt{7}\right)}$$

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}.

$$\frac{13\left(1+\sqrt{7}\right)}{1^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 1. Square \sqrt{7}.

$$\frac{13\left(1+\sqrt{7}\right)}{1-7}$$

Subtract 7 from 1 to get -6.

$$\frac{13\left(1+\sqrt{7}\right)}{-6}$$

Use the distributive property to multiply 13 by 1+\sqrt{7}.

$$\frac{13+13\sqrt{7}}{-6}$$