Solve frac{sqrt{4}+2}{1-sqrt{7}}

$$\frac { \sqrt { 4 } + 2 } { 1 - \sqrt { 7 } }$$

Evaluate

$$\frac{-2\sqrt{7}-2}{3}\approx -2.430500874$$

Solution Steps

Calculate the square root of 4 and get 2.

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

Add 2 and 2 to get 4.

$$\frac{4}{1-\sqrt{7}}$$

Rationalize the denominator of \frac{4}{1-\sqrt{7}} by multiplying numerator and denominator by 1+\sqrt{7}.

$$\frac{4\left(1+\sqrt{7}\right)}{\left(1-\sqrt{7}\right)\left(1+\sqrt{7}\right)}$$

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}.

$$\frac{4\left(1+\sqrt{7}\right)}{1^{2}-\left(\sqrt{7}\right)^{2}}$$

Square 1. Square \sqrt{7}.

$$\frac{4\left(1+\sqrt{7}\right)}{1-7}$$

Subtract 7 from 1 to get -6.

$$\frac{4\left(1+\sqrt{7}\right)}{-6}$$

Divide 4\left(1+\sqrt{7}\right) by -6 to get -\frac{2}{3}\left(1+\sqrt{7}\right).

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

Use the distributive property to multiply -\frac{2}{3} by 1+\sqrt{7}.

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