Solve 60sqrt{1/6}-2sqrt{54}-12sqrt{1/24}

$$60 \sqrt { \frac { 1 } { 6 } } - 2 \sqrt { 54 } - 12 \sqrt { \frac { 1 } { 24 } }$$

Evaluate

$$3\sqrt{6}\approx 7.348469228$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$60\times \frac{\sqrt{1}}{\sqrt{6}}-2\sqrt{54}-12\sqrt{\frac{1}{24}}$$

Calculate the square root of 1 and get 1.

$$60\times \frac{1}{\sqrt{6}}-2\sqrt{54}-12\sqrt{\frac{1}{24}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$60\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}-2\sqrt{54}-12\sqrt{\frac{1}{24}}$$

The square of \sqrt{6} is 6.

$$60\times \frac{\sqrt{6}}{6}-2\sqrt{54}-12\sqrt{\frac{1}{24}}$$

Cancel out 6, the greatest common factor in 60 and 6.

$$10\sqrt{6}-2\sqrt{54}-12\sqrt{\frac{1}{24}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

$$10\sqrt{6}-2\times 3\sqrt{6}-12\sqrt{\frac{1}{24}}$$

Multiply -2 and 3 to get -6.

$$10\sqrt{6}-6\sqrt{6}-12\sqrt{\frac{1}{24}}$$

Combine 10\sqrt{6} and -6\sqrt{6} to get 4\sqrt{6}.

$$4\sqrt{6}-12\sqrt{\frac{1}{24}}$$

Rewrite the square root of the division \sqrt{\frac{1}{24}} as the division of square roots \frac{\sqrt{1}}{\sqrt{24}}.

$$4\sqrt{6}-12\times \frac{\sqrt{1}}{\sqrt{24}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{6}-12\times \frac{1}{\sqrt{24}}$$

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}.

$$4\sqrt{6}-12\times \frac{1}{2\sqrt{6}}$$

Rationalize the denominator of \frac{1}{2\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$4\sqrt{6}-12\times \frac{\sqrt{6}}{2\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$4\sqrt{6}-12\times \frac{\sqrt{6}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$4\sqrt{6}-12\times \frac{\sqrt{6}}{12}$$

Cancel out 12 and 12.

$$4\sqrt{6}-\sqrt{6}$$

Combine 4\sqrt{6} and -\sqrt{6} to get 3\sqrt{6}.

$$3\sqrt{6}$$