Solve sqrt{6}(5+sqrt{18})-frac{24-sqrt{72}}{sqrt{6}}

$$\sqrt { 6 } ( 5 + \sqrt { 18 } ) - \frac { 24 - \sqrt { 72 } } { \sqrt { 6 } }$$

Evaluate

$$\sqrt{6}+8\sqrt{3}\approx 16.305896203$$

Short Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\sqrt{6}\left(5+3\sqrt{2}\right)-\frac{24-\sqrt{72}}{\sqrt{6}}$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$\sqrt{6}\left(5+3\sqrt{2}\right)-\frac{24-6\sqrt{2}}{\sqrt{6}}$$

Rationalize the denominator of \frac{24-6\sqrt{2}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\sqrt{6}\left(5+3\sqrt{2}\right)-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\sqrt{6}\left(5+3\sqrt{2}\right)-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply \sqrt{6} by 5+3\sqrt{2}.

$$5\sqrt{6}+3\sqrt{6}\sqrt{2}-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{6}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$5\sqrt{6}+3\sqrt{2}\sqrt{3}\sqrt{2}-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$5\sqrt{6}+3\times 2\sqrt{3}-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{6}$$

Multiply 3 and 2 to get 6.

$$5\sqrt{6}+6\sqrt{3}-\frac{\left(24-6\sqrt{2}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply 24-6\sqrt{2} by \sqrt{6}.

$$5\sqrt{6}+6\sqrt{3}-\frac{24\sqrt{6}-6\sqrt{2}\sqrt{6}}{6}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$5\sqrt{6}+6\sqrt{3}-\frac{24\sqrt{6}-6\sqrt{2}\sqrt{2}\sqrt{3}}{6}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$5\sqrt{6}+6\sqrt{3}-\frac{24\sqrt{6}-6\times 2\sqrt{3}}{6}$$

Multiply -6 and 2 to get -12.

$$5\sqrt{6}+6\sqrt{3}-\frac{24\sqrt{6}-12\sqrt{3}}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{6}+6\sqrt{3} times \frac{6}{6}.

$$\frac{6\left(5\sqrt{6}+6\sqrt{3}\right)}{6}-\frac{24\sqrt{6}-12\sqrt{3}}{6}$$

Since \frac{6\left(5\sqrt{6}+6\sqrt{3}\right)}{6} and \frac{24\sqrt{6}-12\sqrt{3}}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\left(5\sqrt{6}+6\sqrt{3}\right)-\left(24\sqrt{6}-12\sqrt{3}\right)}{6}$$

Do the multiplications in 6\left(5\sqrt{6}+6\sqrt{3}\right)-\left(24\sqrt{6}-12\sqrt{3}\right).

$$\frac{30\sqrt{6}+36\sqrt{3}-24\sqrt{6}+12\sqrt{3}}{6}$$

Do the calculations in 30\sqrt{6}+36\sqrt{3}-24\sqrt{6}+12\sqrt{3}.

$$\frac{6\sqrt{6}+48\sqrt{3}}{6}$$

Divide each term of 6\sqrt{6}+48\sqrt{3} by 6 to get \sqrt{6}+8\sqrt{3}.

$$\sqrt{6}+8\sqrt{3}$$

Factor

$$\sqrt{6} + 8 \sqrt{3} = 16.305896203$$