Solve sqrt{2/3}-(1/6sqrt{24}-3/2sqrt{12})-1

$$\sqrt{ \frac{ 2 }{ 3 } } -( \frac{ 1 }{ 6 } \sqrt{ 24 } - \frac{ 3 }{ 2 } \sqrt{ 12 } )-1$$

Evaluate

$$3\sqrt{3}-1\approx 4.196152423$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{\sqrt{2}}{\sqrt{3}}-\left(\frac{1}{6}\sqrt{24}-\frac{3}{2}\sqrt{12}\right)-1$$

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

$$\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\left(\frac{1}{6}\sqrt{24}-\frac{3}{2}\sqrt{12}\right)-1$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}\sqrt{3}}{3}-\left(\frac{1}{6}\sqrt{24}-\frac{3}{2}\sqrt{12}\right)-1$$

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

$$\frac{\sqrt{6}}{3}-\left(\frac{1}{6}\sqrt{24}-\frac{3}{2}\sqrt{12}\right)-1$$

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}.

$$\frac{\sqrt{6}}{3}-\left(\frac{1}{6}\times 2\sqrt{6}-\frac{3}{2}\sqrt{12}\right)-1$$

Multiply \frac{1}{6} and 2 to get \frac{2}{6}.

$$\frac{\sqrt{6}}{3}-\left(\frac{2}{6}\sqrt{6}-\frac{3}{2}\sqrt{12}\right)-1$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{\sqrt{6}}{3}-\left(\frac{1}{3}\sqrt{6}-\frac{3}{2}\sqrt{12}\right)-1$$

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}.

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

Cancel out 2 and 2.

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

To find the opposite of \frac{1}{3}\sqrt{6}-3\sqrt{3}, find the opposite of each term.

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

Combine \frac{\sqrt{6}}{3} and -\frac{1}{3}\sqrt{6} to get 0.

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

The opposite of -3\sqrt{3} is 3\sqrt{3}.

$$3\sqrt{3}-1$$

Factor

$$3 \sqrt{3} - 1 = 4.196152423$$