Solve sqrt{3}*(sqrt{12}-4sqrt{1/6})

$$\sqrt{ 3 } \times ( \sqrt{ 12 } -4 \sqrt{ \frac{ 1 }{ 6 } } )$$

Evaluate

$$6-2\sqrt{2}\approx 3.171572875$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\sqrt{3}\left(2\sqrt{3}-4\sqrt{\frac{1}{6}}\right)$$

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

$$\sqrt{3}\left(2\sqrt{3}-4\times \frac{\sqrt{1}}{\sqrt{6}}\right)$$

Calculate the square root of 1 and get 1.

$$\sqrt{3}\left(2\sqrt{3}-4\times \frac{1}{\sqrt{6}}\right)$$

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

$$\sqrt{3}\left(2\sqrt{3}-4\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\right)$$

The square of \sqrt{6} is 6.

$$\sqrt{3}\left(2\sqrt{3}-4\times \frac{\sqrt{6}}{6}\right)$$

Express -4\times \frac{\sqrt{6}}{6} as a single fraction.

$$\sqrt{3}\left(2\sqrt{3}+\frac{-4\sqrt{6}}{6}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3} times \frac{6}{6}.

$$\sqrt{3}\left(\frac{6\times 2\sqrt{3}}{6}+\frac{-4\sqrt{6}}{6}\right)$$

Since \frac{6\times 2\sqrt{3}}{6} and \frac{-4\sqrt{6}}{6} have the same denominator, add them by adding their numerators.

$$\sqrt{3}\times \frac{6\times 2\sqrt{3}-4\sqrt{6}}{6}$$

Do the multiplications in 6\times 2\sqrt{3}-4\sqrt{6}.

$$\sqrt{3}\times \frac{12\sqrt{3}-4\sqrt{6}}{6}$$

Express \sqrt{3}\times \frac{12\sqrt{3}-4\sqrt{6}}{6} as a single fraction.

$$\frac{\sqrt{3}\left(12\sqrt{3}-4\sqrt{6}\right)}{6}$$

Use the distributive property to multiply \sqrt{3} by 12\sqrt{3}-4\sqrt{6}.

$$\frac{12\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{6}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{12\times 3-4\sqrt{3}\sqrt{6}}{6}$$

Multiply 12 and 3 to get 36.

$$\frac{36-4\sqrt{3}\sqrt{6}}{6}$$

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

$$\frac{36-4\sqrt{3}\sqrt{3}\sqrt{2}}{6}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{36-4\times 3\sqrt{2}}{6}$$

Multiply -4 and 3 to get -12.

$$\frac{36-12\sqrt{2}}{6}$$