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

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

Evaluate

$$-\frac{3\sqrt{14}}{4}\approx -2.80624304$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{7}{3}} as the division of square roots \frac{\sqrt{7}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{3\times \frac{\sqrt{21}}{3}\left(-\frac{1}{8}\right)\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

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

$$\frac{\frac{3\left(-1\right)}{8}\times \frac{\sqrt{21}}{3}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Multiply 3 and -1 to get -3.

$$\frac{\frac{-3}{8}\times \frac{\sqrt{21}}{3}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

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

$$\frac{-\frac{3}{8}\times \frac{\sqrt{21}}{3}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Multiply -\frac{3}{8} times \frac{\sqrt{21}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-3\sqrt{21}}{8\times 3}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\frac{-\sqrt{21}}{8}\sqrt{15}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Express \frac{-\sqrt{21}}{8}\sqrt{15} as a single fraction.

$$\frac{\frac{-\sqrt{21}\sqrt{15}}{8}}{\frac{1}{2}}\sqrt{\frac{2}{5}}$$

Divide \frac{-\sqrt{21}\sqrt{15}}{8} by \frac{1}{2} by multiplying \frac{-\sqrt{21}\sqrt{15}}{8} by the reciprocal of \frac{1}{2}.

$$\frac{-\sqrt{21}\sqrt{15}\times 2}{8}\sqrt{\frac{2}{5}}$$

Multiply -1 and 2 to get -2.

$$\frac{-2\sqrt{21}\sqrt{15}}{8}\sqrt{\frac{2}{5}}$$

To multiply \sqrt{21} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{-2\sqrt{315}}{8}\sqrt{\frac{2}{5}}$$

Divide -2\sqrt{315} by 8 to get -\frac{1}{4}\sqrt{315}.

$$-\frac{1}{4}\sqrt{315}\sqrt{\frac{2}{5}}$$

Factor 315=3^{2}\times 35. Rewrite the square root of the product \sqrt{3^{2}\times 35} as the product of square roots \sqrt{3^{2}}\sqrt{35}. Take the square root of 3^{2}.

$$-\frac{1}{4}\times 3\sqrt{35}\sqrt{\frac{2}{5}}$$

Express -\frac{1}{4}\times 3 as a single fraction.

$$\frac{-3}{4}\sqrt{35}\sqrt{\frac{2}{5}}$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$-\frac{3}{4}\sqrt{35}\sqrt{\frac{2}{5}}$$

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

$$-\frac{3}{4}\sqrt{35}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

$$-\frac{3}{4}\sqrt{35}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$-\frac{3}{4}\sqrt{35}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

$$-\frac{3}{4}\sqrt{35}\times \frac{\sqrt{10}}{5}$$

Multiply -\frac{3}{4} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-3\sqrt{10}}{4\times 5}\sqrt{35}$$

Multiply 4 and 5 to get 20.

$$\frac{-3\sqrt{10}}{20}\sqrt{35}$$

Express \frac{-3\sqrt{10}}{20}\sqrt{35} as a single fraction.

$$\frac{-3\sqrt{10}\sqrt{35}}{20}$$

To multiply \sqrt{10} and \sqrt{35}, multiply the numbers under the square root.

$$\frac{-3\sqrt{350}}{20}$$

Factor 350=5^{2}\times 14. Rewrite the square root of the product \sqrt{5^{2}\times 14} as the product of square roots \sqrt{5^{2}}\sqrt{14}. Take the square root of 5^{2}.

$$\frac{-3\times 5\sqrt{14}}{20}$$

Multiply -3 and 5 to get -15.

$$\frac{-15\sqrt{14}}{20}$$

Divide -15\sqrt{14} by 20 to get -\frac{3}{4}\sqrt{14}.

$$-\frac{3}{4}\sqrt{14}$$