Solve (2sqrt{30}-sqrt{24})*sqrt{1/6}

$$( 2 \sqrt { 30 } - \sqrt { 24 } ) \cdot \sqrt { \frac { 1 } { 6 } }$$

Evaluate

$$2\left(\sqrt{5}-1\right)\approx 2.472135955$$

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}.

$$\left(2\sqrt{30}-2\sqrt{6}\right)\sqrt{\frac{1}{6}}$$

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

$$\left(2\sqrt{30}-2\sqrt{6}\right)\times \frac{\sqrt{1}}{\sqrt{6}}$$

Calculate the square root of 1 and get 1.

$$\left(2\sqrt{30}-2\sqrt{6}\right)\times \frac{1}{\sqrt{6}}$$

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

$$\left(2\sqrt{30}-2\sqrt{6}\right)\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\left(2\sqrt{30}-2\sqrt{6}\right)\times \frac{\sqrt{6}}{6}$$

Express \left(2\sqrt{30}-2\sqrt{6}\right)\times \frac{\sqrt{6}}{6} as a single fraction.

$$\frac{\left(2\sqrt{30}-2\sqrt{6}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply 2\sqrt{30}-2\sqrt{6} by \sqrt{6}.

$$\frac{2\sqrt{30}\sqrt{6}-2\left(\sqrt{6}\right)^{2}}{6}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

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

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

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

Multiply 2 and 6 to get 12.

$$\frac{12\sqrt{5}-2\left(\sqrt{6}\right)^{2}}{6}$$

The square of \sqrt{6} is 6.

$$\frac{12\sqrt{5}-2\times 6}{6}$$

Multiply -2 and 6 to get -12.

$$\frac{12\sqrt{5}-12}{6}$$