Solve 3sqrt{3frac{1/2}-1/8sqrt{14/7}}{1/2}sqrt{5frac{1}{2}}

$$\frac{ 3 \sqrt{ 3 \frac{ 1 }{ 2 } } - \frac{ 1 }{ 8 } \sqrt{ 1 \frac{ 4 }{ 7 } } }{ \frac{ 1 }{ 2 } } \sqrt{ 5 \frac{ 1 }{ 2 } }$$

Evaluate

$$\frac{6\sqrt{77}}{2}-\frac{11\sqrt{14}}{56}\approx 25.589924747$$

Short Solution Steps

Multiply 3 and 2 to get 6.

$$\frac{3\sqrt{\frac{6+1}{2}}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Add 6 and 1 to get 7.

$$\frac{3\sqrt{\frac{7}{2}}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

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

$$\frac{3\times \frac{\sqrt{7}}{\sqrt{2}}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

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

$$\frac{3\times \frac{\sqrt{7}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3\times \frac{\sqrt{7}\sqrt{2}}{2}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

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

$$\frac{3\times \frac{\sqrt{14}}{2}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Express 3\times \frac{\sqrt{14}}{2} as a single fraction.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 1 and 7 to get 7.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\sqrt{\frac{7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Add 7 and 4 to get 11.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\sqrt{\frac{11}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{11}{7}} as the division of square roots \frac{\sqrt{11}}{\sqrt{7}}.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\times \frac{\sqrt{11}}{\sqrt{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\times \frac{\sqrt{11}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\times \frac{\sqrt{11}\sqrt{7}}{7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

To multiply \sqrt{11} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{1}{8}\times \frac{\sqrt{77}}{7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply \frac{1}{8} times \frac{\sqrt{77}}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{\sqrt{77}}{8\times 7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 8 and 7 to get 56.

$$\frac{\frac{3\sqrt{14}}{2}-\frac{\sqrt{77}}{56}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 56 is 56. Multiply \frac{3\sqrt{14}}{2} times \frac{28}{28}.

$$\frac{\frac{28\times 3\sqrt{14}}{56}-\frac{\sqrt{77}}{56}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Since \frac{28\times 3\sqrt{14}}{56} and \frac{\sqrt{77}}{56} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{28\times 3\sqrt{14}-\sqrt{77}}{56}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Do the multiplications in 28\times 3\sqrt{14}-\sqrt{77}.

$$\frac{\frac{84\sqrt{14}-\sqrt{77}}{56}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Divide \frac{84\sqrt{14}-\sqrt{77}}{56} by \frac{1}{2} by multiplying \frac{84\sqrt{14}-\sqrt{77}}{56} by the reciprocal of \frac{1}{2}.

$$\frac{\left(84\sqrt{14}-\sqrt{77}\right)\times 2}{56}\sqrt{\frac{5\times 2+1}{2}}$$

Divide \left(84\sqrt{14}-\sqrt{77}\right)\times 2 by 56 to get \left(84\sqrt{14}-\sqrt{77}\right)\times \frac{1}{28}.

$$\left(84\sqrt{14}-\sqrt{77}\right)\times \frac{1}{28}\sqrt{\frac{5\times 2+1}{2}}$$

Use the distributive property to multiply 84\sqrt{14}-\sqrt{77} by \frac{1}{28}.

$$\left(84\sqrt{14}\times \frac{1}{28}-\sqrt{77}\times \frac{1}{28}\right)\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 84 and \frac{1}{28} to get \frac{84}{28}.

$$\left(\frac{84}{28}\sqrt{14}-\sqrt{77}\times \frac{1}{28}\right)\sqrt{\frac{5\times 2+1}{2}}$$

Divide 84 by 28 to get 3.

$$\left(3\sqrt{14}-\sqrt{77}\times \frac{1}{28}\right)\sqrt{\frac{5\times 2+1}{2}}$$

Multiply -1 and \frac{1}{28} to get -\frac{1}{28}.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 5 and 2 to get 10.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\sqrt{\frac{10+1}{2}}$$

Add 10 and 1 to get 11.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\sqrt{\frac{11}{2}}$$

Rewrite the square root of the division \sqrt{\frac{11}{2}} as the division of square roots \frac{\sqrt{11}}{\sqrt{2}}.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\times \frac{\sqrt{11}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\times \frac{\sqrt{11}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\times \frac{\sqrt{11}\sqrt{2}}{2}$$

To multiply \sqrt{11} and \sqrt{2}, multiply the numbers under the square root.

$$\left(3\sqrt{14}-\frac{1}{28}\sqrt{77}\right)\times \frac{\sqrt{22}}{2}$$

Use the distributive property to multiply 3\sqrt{14}-\frac{1}{28}\sqrt{77} by \frac{\sqrt{22}}{2}.

$$3\sqrt{14}\times \frac{\sqrt{22}}{2}-\frac{1}{28}\sqrt{77}\times \frac{\sqrt{22}}{2}$$

Express 3\times \frac{\sqrt{22}}{2} as a single fraction.

$$\frac{3\sqrt{22}}{2}\sqrt{14}-\frac{1}{28}\sqrt{77}\times \frac{\sqrt{22}}{2}$$

Multiply -\frac{1}{28} times \frac{\sqrt{22}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{22}}{2}\sqrt{14}+\frac{-\sqrt{22}}{28\times 2}\sqrt{77}$$

Express \frac{3\sqrt{22}}{2}\sqrt{14} as a single fraction.

$$\frac{3\sqrt{22}\sqrt{14}}{2}+\frac{-\sqrt{22}}{28\times 2}\sqrt{77}$$

Multiply 28 and 2 to get 56.

$$\frac{3\sqrt{22}\sqrt{14}}{2}+\frac{-\sqrt{22}}{56}\sqrt{77}$$

Express \frac{-\sqrt{22}}{56}\sqrt{77} as a single fraction.

$$\frac{3\sqrt{22}\sqrt{14}}{2}+\frac{-\sqrt{22}\sqrt{77}}{56}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 56 is 56. Multiply \frac{3\sqrt{22}\sqrt{14}}{2} times \frac{28}{28}.

$$\frac{28\times 3\sqrt{22}\sqrt{14}}{56}+\frac{-\sqrt{22}\sqrt{77}}{56}$$

Since \frac{28\times 3\sqrt{22}\sqrt{14}}{56} and \frac{-\sqrt{22}\sqrt{77}}{56} have the same denominator, add them by adding their numerators.

$$\frac{28\times 3\sqrt{22}\sqrt{14}-\sqrt{22}\sqrt{77}}{56}$$

Do the multiplications in 28\times 3\sqrt{22}\sqrt{14}-\sqrt{22}\sqrt{77}.

$$\frac{168\sqrt{77}-11\sqrt{14}}{56}$$

Factor

$$\frac{\frac{1}{7} \sqrt{22} {(84 \sqrt{14} - \sqrt{77})}}{8} = 25.589924746917198$$