Solve (sqrt{8/27}-5sqrt{3})*6

$$( \sqrt { \frac { 8 } { 27 } } - 5 \sqrt { 3 } ) \times 6$$

Evaluate

$$\frac{4\sqrt{6}}{3}-30\sqrt{3}\approx -48.695537903$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{8}{27}} as the division of square roots \frac{\sqrt{8}}{\sqrt{27}}.

$$6\left(\frac{\sqrt{8}}{\sqrt{27}}-5\sqrt{3}\right)$$

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}.

$$6\left(\frac{2\sqrt{2}}{\sqrt{27}}-5\sqrt{3}\right)$$

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}.

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

Rationalize the denominator of \frac{2\sqrt{2}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

Multiply 3 and 3 to get 9.

$$6\left(\frac{2\sqrt{6}}{9}-5\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5\sqrt{3} times \frac{9}{9}.

$$6\left(\frac{2\sqrt{6}}{9}+\frac{9\left(-5\right)\sqrt{3}}{9}\right)$$

Since \frac{2\sqrt{6}}{9} and \frac{9\left(-5\right)\sqrt{3}}{9} have the same denominator, add them by adding their numerators.

$$6\times \frac{2\sqrt{6}+9\left(-5\right)\sqrt{3}}{9}$$

Do the multiplications in 2\sqrt{6}+9\left(-5\right)\sqrt{3}.

$$6\times \frac{2\sqrt{6}-45\sqrt{3}}{9}$$

Express 6\times \frac{2\sqrt{6}-45\sqrt{3}}{9} as a single fraction.

$$\frac{6\left(2\sqrt{6}-45\sqrt{3}\right)}{9}$$

Use the distributive property to multiply 6 by 2\sqrt{6}-45\sqrt{3}.

$$\frac{12\sqrt{6}-270\sqrt{3}}{9}$$