Solve 3sqrt{2frac{2/3}}{1/2}sqrt{2/7}-1/8sqrt{4}

$$\frac{ 3 \sqrt{ 2 \frac{ 2 }{ 3 } } }{ \frac{ 1 }{ 2 } } \sqrt{ \frac{ 2 }{ 7 } } - \frac{ 1 }{ 8 } \sqrt{ 4 }$$

Evaluate

$$\frac{8\sqrt{21}}{7}-\frac{1}{4}\approx 4.987229366$$

Short Solution Steps

Multiply 2 and 3 to get 6.

$$\frac{3\sqrt{\frac{6+2}{3}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Add 6 and 2 to get 8.

$$\frac{3\sqrt{\frac{8}{3}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{3\times \frac{\sqrt{8}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

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}.

$$\frac{3\times \frac{2\sqrt{2}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

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

$$\frac{3\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

The square of \sqrt{3} is 3.

$$\frac{3\times \frac{2\sqrt{2}\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

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

$$\frac{3\times \frac{2\sqrt{6}}{3}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Cancel out 3 and 3.

$$\frac{2\sqrt{6}}{\frac{1}{2}}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Divide 2\sqrt{6} by \frac{1}{2} by multiplying 2\sqrt{6} by the reciprocal of \frac{1}{2}.

$$2\sqrt{6}\times 2\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Multiply 2 and 2 to get 4.

$$4\sqrt{6}\sqrt{\frac{2}{7}}-\frac{1}{8}\sqrt{4}$$

Rewrite the square root of the division \sqrt{\frac{2}{7}} as the division of square roots \frac{\sqrt{2}}{\sqrt{7}}.

$$4\sqrt{6}\times \frac{\sqrt{2}}{\sqrt{7}}-\frac{1}{8}\sqrt{4}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$4\sqrt{6}\times \frac{\sqrt{2}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}-\frac{1}{8}\sqrt{4}$$

The square of \sqrt{7} is 7.

$$4\sqrt{6}\times \frac{\sqrt{2}\sqrt{7}}{7}-\frac{1}{8}\sqrt{4}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$4\sqrt{6}\times \frac{\sqrt{14}}{7}-\frac{1}{8}\sqrt{4}$$

Express 4\times \frac{\sqrt{14}}{7} as a single fraction.

$$\frac{4\sqrt{14}}{7}\sqrt{6}-\frac{1}{8}\sqrt{4}$$

Calculate the square root of 4 and get 2.

$$\frac{4\sqrt{14}}{7}\sqrt{6}-\frac{1}{8}\times 2$$

Express -\frac{1}{8}\times 2 as a single fraction.

$$\frac{4\sqrt{14}}{7}\sqrt{6}+\frac{-2}{8}$$

Reduce the fraction \frac{-2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{4\sqrt{14}}{7}\sqrt{6}-\frac{1}{4}$$

Express \frac{4\sqrt{14}}{7}\sqrt{6} as a single fraction.

$$\frac{4\sqrt{14}\sqrt{6}}{7}-\frac{1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 4 is 28. Multiply \frac{4\sqrt{14}\sqrt{6}}{7} times \frac{4}{4}. Multiply \frac{1}{4} times \frac{7}{7}.

$$\frac{4\times 4\sqrt{14}\sqrt{6}}{28}-\frac{7}{28}$$

Since \frac{4\times 4\sqrt{14}\sqrt{6}}{28} and \frac{7}{28} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\times 4\sqrt{14}\sqrt{6}-7}{28}$$

Do the multiplications in 4\times 4\sqrt{14}\sqrt{6}-7.

$$\frac{32\sqrt{21}-7}{28}$$

Factor

$$\frac{32 \sqrt{21} - 7}{28} = 4.987229365663817$$