Solve 16/sqrt{6-sqrt{2}}+frac{2(sqrt{6}-sqrt{2})}{8}

$$\frac { 16 } { \sqrt { 6 } - \sqrt { 2 } } + \frac { 2 ( \sqrt { 6 } - \sqrt { 2 } ) } { 8 }$$

Evaluate

$$\frac{15\sqrt{2}+17\sqrt{6}}{4}\approx 15.713632266$$

Solution Steps

Rationalize the denominator of \frac{16}{\sqrt{6}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{2}.

$$\frac{16\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}+\frac{2\left(\sqrt{6}-\sqrt{2}\right)}{8}$$

Consider \left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{16\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}+\frac{2\left(\sqrt{6}-\sqrt{2}\right)}{8}$$

Square \sqrt{6}. Square \sqrt{2}.

$$\frac{16\left(\sqrt{6}+\sqrt{2}\right)}{6-2}+\frac{2\left(\sqrt{6}-\sqrt{2}\right)}{8}$$

Subtract 2 from 6 to get 4.

$$\frac{16\left(\sqrt{6}+\sqrt{2}\right)}{4}+\frac{2\left(\sqrt{6}-\sqrt{2}\right)}{8}$$

Divide 16\left(\sqrt{6}+\sqrt{2}\right) by 4 to get 4\left(\sqrt{6}+\sqrt{2}\right).

$$4\left(\sqrt{6}+\sqrt{2}\right)+\frac{2\left(\sqrt{6}-\sqrt{2}\right)}{8}$$

Divide 2\left(\sqrt{6}-\sqrt{2}\right) by 8 to get \frac{1}{4}\left(\sqrt{6}-\sqrt{2}\right).

$$4\left(\sqrt{6}+\sqrt{2}\right)+\frac{1}{4}\left(\sqrt{6}-\sqrt{2}\right)$$

Use the distributive property to multiply 4 by \sqrt{6}+\sqrt{2}.

$$4\sqrt{6}+4\sqrt{2}+\frac{1}{4}\left(\sqrt{6}-\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{4} by \sqrt{6}-\sqrt{2}.

$$4\sqrt{6}+4\sqrt{2}+\frac{1}{4}\sqrt{6}+\frac{1}{4}\left(-1\right)\sqrt{2}$$

Multiply \frac{1}{4} and -1 to get -\frac{1}{4}.

$$4\sqrt{6}+4\sqrt{2}+\frac{1}{4}\sqrt{6}-\frac{1}{4}\sqrt{2}$$

Combine 4\sqrt{6} and \frac{1}{4}\sqrt{6} to get \frac{17}{4}\sqrt{6}.

$$\frac{17}{4}\sqrt{6}+4\sqrt{2}-\frac{1}{4}\sqrt{2}$$

Combine 4\sqrt{2} and -\frac{1}{4}\sqrt{2} to get \frac{15}{4}\sqrt{2}.

$$\frac{17}{4}\sqrt{6}+\frac{15}{4}\sqrt{2}$$

Factor

$$\frac{15 \sqrt{2} + 17 \sqrt{6}}{4} = 15.713632265727613$$