Solve 3sqrt{3}-(sqrt{12}+sqrt{1/27})

$$3 \sqrt { 3 } - ( \sqrt { 12 } + \sqrt { \frac { 1 } { 27 } } )$$

Evaluate

$$\frac{8\sqrt{3}}{9}\approx 1.539600718$$

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}.

$$3\sqrt{3}-\left(2\sqrt{3}+\sqrt{\frac{1}{27}}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{27}} as the division of square roots \frac{\sqrt{1}}{\sqrt{27}}.

$$3\sqrt{3}-\left(2\sqrt{3}+\frac{\sqrt{1}}{\sqrt{27}}\right)$$

Calculate the square root of 1 and get 1.

$$3\sqrt{3}-\left(2\sqrt{3}+\frac{1}{\sqrt{27}}\right)$$

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}.

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

Rationalize the denominator of \frac{1}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

$$3\sqrt{3}-\left(2\sqrt{3}+\frac{\sqrt{3}}{9}\right)$$

Combine 2\sqrt{3} and \frac{\sqrt{3}}{9} to get \frac{19}{9}\sqrt{3}.

$$3\sqrt{3}-\frac{19}{9}\sqrt{3}$$

Combine 3\sqrt{3} and -\frac{19}{9}\sqrt{3} to get \frac{8}{9}\sqrt{3}.

$$\frac{8}{9}\sqrt{3}$$