Solve 2/2+sqrt{14}

$$\frac { 2 } { 2 + \sqrt { 14 } }$$

Evaluate

$$\frac{\sqrt{14}-2}{5}\approx 0.348331477$$

Solution Steps

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

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

Consider \left(2+\sqrt{14}\right)\left(2-\sqrt{14}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 2. Square \sqrt{14}.

$$\frac{2\left(2-\sqrt{14}\right)}{4-14}$$

Subtract 14 from 4 to get -10.

$$\frac{2\left(2-\sqrt{14}\right)}{-10}$$

Divide 2\left(2-\sqrt{14}\right) by -10 to get -\frac{1}{5}\left(2-\sqrt{14}\right).

$$-\frac{1}{5}\left(2-\sqrt{14}\right)$$

Use the distributive property to multiply -\frac{1}{5} by 2-\sqrt{14}.

$$-\frac{1}{5}\times 2-\frac{1}{5}\left(-1\right)\sqrt{14}$$

Express -\frac{1}{5}\times 2 as a single fraction.

$$\frac{-2}{5}-\frac{1}{5}\left(-1\right)\sqrt{14}$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$-\frac{2}{5}-\frac{1}{5}\left(-1\right)\sqrt{14}$$

Multiply -\frac{1}{5} and -1 to get \frac{1}{5}.

$$-\frac{2}{5}+\frac{1}{5}\sqrt{14}$$