Solve sqrt{31/2}*(-8sqrt{14/7})div1/2sqrt{51/2}

$$\sqrt { 3 \frac { 1 } { 2 } } \times ( - 8 \sqrt { 1 \frac { 4 } { 7 } } ) \div \frac { 1 } { 2 } \sqrt { 5 \frac { 1 } { 2 } }$$

Evaluate

$$-88$$

Solution Steps

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{\frac{6+1}{2}}\left(-8\right)\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Add 6 and 1 to get 7.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

$$\frac{\frac{\sqrt{14}}{2}\left(-8\right)\sqrt{\frac{1\times 7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 1 and 7 to get 7.

$$\frac{\frac{\sqrt{14}}{2}\left(-8\right)\sqrt{\frac{7+4}{7}}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Add 7 and 4 to get 11.

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

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

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

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

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

The square of \sqrt{7} is 7.

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

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

$$\frac{\frac{\sqrt{14}}{2}\left(-8\right)\times \frac{\sqrt{77}}{7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Cancel out 2, the greatest common factor in 8 and 2.

$$\frac{-4\sqrt{14}\times \frac{\sqrt{77}}{7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Express -4\sqrt{14}\times \frac{\sqrt{77}}{7} as a single fraction.

$$\frac{\frac{-4\sqrt{14}\sqrt{77}}{7}}{\frac{1}{2}}\sqrt{\frac{5\times 2+1}{2}}$$

Divide \frac{-4\sqrt{14}\sqrt{77}}{7} by \frac{1}{2} by multiplying \frac{-4\sqrt{14}\sqrt{77}}{7} by the reciprocal of \frac{1}{2}.

$$\frac{-4\sqrt{14}\sqrt{77}\times 2}{7}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply -4 and 2 to get -8.

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

To multiply \sqrt{14} and \sqrt{77}, multiply the numbers under the square root.

$$\frac{-8\sqrt{1078}}{7}\sqrt{\frac{5\times 2+1}{2}}$$

Multiply 5 and 2 to get 10.

$$\frac{-8\sqrt{1078}}{7}\sqrt{\frac{10+1}{2}}$$

Add 10 and 1 to get 11.

$$\frac{-8\sqrt{1078}}{7}\sqrt{\frac{11}{2}}$$

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

$$\frac{-8\sqrt{1078}}{7}\times \frac{\sqrt{11}}{\sqrt{2}}$$

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

$$\frac{-8\sqrt{1078}}{7}\times \frac{\sqrt{11}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-8\sqrt{1078}}{7}\times \frac{\sqrt{11}\sqrt{2}}{2}$$

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

$$\frac{-8\sqrt{1078}}{7}\times \frac{\sqrt{22}}{2}$$

Multiply \frac{-8\sqrt{1078}}{7} times \frac{\sqrt{22}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-8\sqrt{1078}\sqrt{22}}{7\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-4\sqrt{22}\sqrt{1078}}{7}$$

Factor 1078=22\times 49. Rewrite the square root of the product \sqrt{22\times 49} as the product of square roots \sqrt{22}\sqrt{49}.

$$\frac{-4\sqrt{22}\sqrt{22}\sqrt{49}}{7}$$

Multiply \sqrt{22} and \sqrt{22} to get 22.

$$\frac{-4\times 22\sqrt{49}}{7}$$

Multiply -4 and 22 to get -88.

$$\frac{-88\sqrt{49}}{7}$$

Calculate the square root of 49 and get 7.

$$\frac{-88\times 7}{7}$$

Multiply -88 and 7 to get -616.

$$\frac{-616}{7}$$

Divide -616 by 7 to get -88.

$$-88$$

Factor

$$-88$$