Solve frac{sqrt{6}+sqrt{8}}{sqrt{23}}

$$\frac { \sqrt { 6 } + \sqrt { 8 } } { \sqrt { 23 } }$$

Evaluate

$$\frac{\sqrt{138}+2\sqrt{46}}{23}\approx 1.100521743$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\sqrt{6}+2\sqrt{2}}{\sqrt{23}}$$

Rationalize the denominator of \frac{\sqrt{6}+2\sqrt{2}}{\sqrt{23}} by multiplying numerator and denominator by \sqrt{23}.

$$\frac{\left(\sqrt{6}+2\sqrt{2}\right)\sqrt{23}}{\left(\sqrt{23}\right)^{2}}$$

The square of \sqrt{23} is 23.

$$\frac{\left(\sqrt{6}+2\sqrt{2}\right)\sqrt{23}}{23}$$

Use the distributive property to multiply \sqrt{6}+2\sqrt{2} by \sqrt{23}.

$$\frac{\sqrt{6}\sqrt{23}+2\sqrt{2}\sqrt{23}}{23}$$

To multiply \sqrt{6} and \sqrt{23}, multiply the numbers under the square root.

$$\frac{\sqrt{138}+2\sqrt{2}\sqrt{23}}{23}$$

To multiply \sqrt{2} and \sqrt{23}, multiply the numbers under the square root.

$$\frac{\sqrt{138}+2\sqrt{46}}{23}$$