Solve frac{sqrt{24}+sqrt{216}}{sqrt{6}}+5

$$\frac { \sqrt { 24 } + \sqrt { 216 } } { \sqrt { 6 } } + 5$$

Evaluate

$$13$$

Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\frac{2\sqrt{6}+\sqrt{216}}{\sqrt{6}}+5$$

Factor 216=6^{2}\times 6. Rewrite the square root of the product \sqrt{6^{2}\times 6} as the product of square roots \sqrt{6^{2}}\sqrt{6}. Take the square root of 6^{2}.

$$\frac{2\sqrt{6}+6\sqrt{6}}{\sqrt{6}}+5$$

Combine 2\sqrt{6} and 6\sqrt{6} to get 8\sqrt{6}.

$$\frac{8\sqrt{6}}{\sqrt{6}}+5$$

Rationalize the denominator of \frac{8\sqrt{6}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{8\sqrt{6}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}+5$$

The square of \sqrt{6} is 6.

$$\frac{8\sqrt{6}\sqrt{6}}{6}+5$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{8\times 6}{6}+5$$

Multiply 8 and 6 to get 48.

$$\frac{48}{6}+5$$

Divide 48 by 6 to get 8.

$$8+5$$

Add 8 and 5 to get 13.

$$13$$

Factor

$$13$$