Solve 3sqrt{7/2}*(-1/8sqrt{15})div1/2sqrt{2/5}

$$3 \sqrt{ \frac{ 7 }{ 2 } } \times (- \frac{ 1 }{ 8 } \sqrt{ 15 } ) \div \frac{ 1 }{ 2 } \sqrt{ \frac{ 2 }{ 5 } }$$

Evaluate

$$-\frac{3\sqrt{21}}{4}\approx -3.436931771$$

Solution Steps

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

$$\frac{3\times \frac{\sqrt{7}}{\sqrt{2}}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

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

$$\frac{3\times \frac{\sqrt{7}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

The square of \sqrt{2} is 2.

$$\frac{3\times \frac{\sqrt{7}\sqrt{2}}{2}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

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

$$\frac{3\times \frac{\sqrt{14}}{2}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Express 3\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{\frac{3\left(-1\right)}{8}\times \frac{\sqrt{14}}{2}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Multiply 3 and -1 to get -3.

$$\frac{\frac{-3}{8}\times \frac{\sqrt{14}}{2}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Fraction \frac{-3}{8} can be rewritten as -\frac{3}{8} by extracting the negative sign.

$$\frac{-\frac{3}{8}\times \frac{\sqrt{14}}{2}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Multiply -\frac{3}{8} times \frac{\sqrt{14}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-3\sqrt{14}}{8\times 2}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Express \frac{-3\sqrt{14}}{8\times 2}\sqrt{15} as a single fraction.

$$\frac{\frac{-3\sqrt{14}\sqrt{15}}{8\times 2}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Divide \frac{-3\sqrt{14}\sqrt{15}}{8\times 2} by \frac{1}{2} by multiplying \frac{-3\sqrt{14}\sqrt{15}}{8\times 2} by the reciprocal of \frac{1}{2}.

$$\frac{-3\sqrt{14}\sqrt{15}\times 2}{8\times 2}\sqrt{\frac{2}{5}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-3\sqrt{14}\sqrt{15}}{2\times 4}\sqrt{\frac{2}{5}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{3\sqrt{14}\sqrt{15}}{-2\times 4}\sqrt{\frac{2}{5}}$$

To multiply \sqrt{14} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{3\sqrt{210}}{-2\times 4}\sqrt{\frac{2}{5}}$$

Multiply -2 and 4 to get -8.

$$\frac{3\sqrt{210}}{-8}\sqrt{\frac{2}{5}}$$

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

$$\frac{3\sqrt{210}}{-8}\times \frac{\sqrt{2}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{3\sqrt{210}}{-8}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{210}}{-8}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{3\sqrt{210}}{-8}\times \frac{\sqrt{10}}{5}$$

Multiply \frac{3\sqrt{210}}{-8} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{210}\sqrt{10}}{-8\times 5}$$

Factor 210=10\times 21. Rewrite the square root of the product \sqrt{10\times 21} as the product of square roots \sqrt{10}\sqrt{21}.

$$\frac{3\sqrt{10}\sqrt{21}\sqrt{10}}{-8\times 5}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{3\times 10\sqrt{21}}{-8\times 5}$$

Multiply 3 and 10 to get 30.

$$\frac{30\sqrt{21}}{-8\times 5}$$

Multiply -8 and 5 to get -40.

$$\frac{30\sqrt{21}}{-40}$$

Divide 30\sqrt{21} by -40 to get -\frac{3}{4}\sqrt{21}.

$$-\frac{3}{4}\sqrt{21}$$