Solve (sqrt{8}-2sqrt{12}+sqrt{18})*sqrt{1/6}

$$( \sqrt { 8 } - 2 \sqrt { 12 } + \sqrt { 18 } ) \times \sqrt { \frac { 1 } { 6 } }$$

Evaluate

$$\frac{5\sqrt{3}}{3}-2\sqrt{2}\approx 0.058324221$$

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}.

$$\left(2\sqrt{2}-2\sqrt{12}+\sqrt{18}\right)\sqrt{\frac{1}{6}}$$

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}.

$$\left(2\sqrt{2}-2\times 2\sqrt{3}+\sqrt{18}\right)\sqrt{\frac{1}{6}}$$

Multiply -2 and 2 to get -4.

$$\left(2\sqrt{2}-4\sqrt{3}+\sqrt{18}\right)\sqrt{\frac{1}{6}}$$

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}.

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

Combine 2\sqrt{2} and 3\sqrt{2} to get 5\sqrt{2}.

$$\left(5\sqrt{2}-4\sqrt{3}\right)\sqrt{\frac{1}{6}}$$

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

$$\left(5\sqrt{2}-4\sqrt{3}\right)\times \frac{\sqrt{1}}{\sqrt{6}}$$

Calculate the square root of 1 and get 1.

$$\left(5\sqrt{2}-4\sqrt{3}\right)\times \frac{1}{\sqrt{6}}$$

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

$$\left(5\sqrt{2}-4\sqrt{3}\right)\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\left(5\sqrt{2}-4\sqrt{3}\right)\times \frac{\sqrt{6}}{6}$$

Express \left(5\sqrt{2}-4\sqrt{3}\right)\times \frac{\sqrt{6}}{6} as a single fraction.

$$\frac{\left(5\sqrt{2}-4\sqrt{3}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply 5\sqrt{2}-4\sqrt{3} by \sqrt{6}.

$$\frac{5\sqrt{2}\sqrt{6}-4\sqrt{3}\sqrt{6}}{6}$$

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}.

$$\frac{5\sqrt{2}\sqrt{2}\sqrt{3}-4\sqrt{3}\sqrt{6}}{6}$$

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

$$\frac{5\times 2\sqrt{3}-4\sqrt{3}\sqrt{6}}{6}$$

Multiply 5 and 2 to get 10.

$$\frac{10\sqrt{3}-4\sqrt{3}\sqrt{6}}{6}$$

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}.

$$\frac{10\sqrt{3}-4\sqrt{3}\sqrt{3}\sqrt{2}}{6}$$

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

$$\frac{10\sqrt{3}-4\times 3\sqrt{2}}{6}$$

Multiply -4 and 3 to get -12.

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