Solve frac{sqrt{14}+2}{1-sqrt{7}}

$$\frac{ \sqrt{ 14 } +2 }{ 1- \sqrt{ 7 } }$$

Evaluate

$$-\frac{\sqrt{7}}{3}-\frac{\sqrt{14}}{6}-\frac{7\sqrt{2}}{6}-\frac{1}{3}\approx -3.488775824$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{14}+2}{1-\sqrt{7}} by multiplying numerator and denominator by 1+\sqrt{7}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{\left(1-\sqrt{7}\right)\left(1+\sqrt{7}\right)}$$

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}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{1^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 1. Square \sqrt{7}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{1-7}$$

Subtract 7 from 1 to get -6.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{7}\right)}{-6}$$

Apply the distributive property by multiplying each term of \sqrt{14}+2 by each term of 1+\sqrt{7}.

$$\frac{\sqrt{14}+\sqrt{14}\sqrt{7}+2+2\sqrt{7}}{-6}$$

Factor 14=7\times 2. Rewrite the square root of the product \sqrt{7\times 2} as the product of square roots \sqrt{7}\sqrt{2}.

$$\frac{\sqrt{14}+\sqrt{7}\sqrt{2}\sqrt{7}+2+2\sqrt{7}}{-6}$$

Multiply \sqrt{7} and \sqrt{7} to get 7.

$$\frac{\sqrt{14}+7\sqrt{2}+2+2\sqrt{7}}{-6}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{14}-7\sqrt{2}-2-2\sqrt{7}}{6}$$