Solve (1/3sqrt{27}-4sqrt{1/5})-2(sqrt{3/4}-sqrt{12})

$$( \frac { 1 } { 3 } \sqrt { 27 } - 4 \sqrt { \frac { 1 } { 5 } } ) - 2 ( \sqrt { \frac { 3 } { 4 } } - \sqrt { 12 } )$$

Evaluate

$$\frac{4\sqrt{5}\left(\sqrt{15}-1\right)}{5}\approx 5.139348848$$

Short Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{1}{3}\times 3\sqrt{3}-4\sqrt{\frac{1}{5}}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Cancel out 3 and 3.

$$\sqrt{3}-4\sqrt{\frac{1}{5}}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$\sqrt{3}-4\times \frac{\sqrt{1}}{\sqrt{5}}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Calculate the square root of 1 and get 1.

$$\sqrt{3}-4\times \frac{1}{\sqrt{5}}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\sqrt{3}-4\times \frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

The square of \sqrt{5} is 5.

$$\sqrt{3}-4\times \frac{\sqrt{5}}{5}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Express -4\times \frac{\sqrt{5}}{5} as a single fraction.

$$\sqrt{3}+\frac{-4\sqrt{5}}{5}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

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

$$\frac{5\sqrt{3}}{5}+\frac{-4\sqrt{5}}{5}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Since \frac{5\sqrt{3}}{5} and \frac{-4\sqrt{5}}{5} have the same denominator, add them by adding their numerators.

$$\frac{5\sqrt{3}-4\sqrt{5}}{5}-2\left(\sqrt{\frac{3}{4}}-\sqrt{12}\right)$$

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$\frac{5\sqrt{3}-4\sqrt{5}}{5}-2\left(\frac{\sqrt{3}}{\sqrt{4}}-\sqrt{12}\right)$$

Calculate the square root of 4 and get 2.

$$\frac{5\sqrt{3}-4\sqrt{5}}{5}-2\left(\frac{\sqrt{3}}{2}-\sqrt{12}\right)$$

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}.

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

Combine \frac{\sqrt{3}}{2} and -2\sqrt{3} to get -\frac{3}{2}\sqrt{3}.

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

Cancel out 2 and 2.

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

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

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

Since \frac{5\sqrt{3}-4\sqrt{5}}{5} and \frac{5\left(-3\right)\sqrt{3}}{5} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\sqrt{3}-4\sqrt{5}-5\left(-3\right)\sqrt{3}.

$$\frac{5\sqrt{3}-4\sqrt{5}+15\sqrt{3}}{5}$$

Do the calculations in 5\sqrt{3}-4\sqrt{5}+15\sqrt{3}.

$$\frac{20\sqrt{3}-4\sqrt{5}}{5}$$

Factor

$$\frac{4 \sqrt{5} {(\sqrt{3} \sqrt{5} - 1)}}{5} = 5.139348848275678$$