Solve 4/3sqrt{42/7}*(-3/2sqrt{22/3})divfrac{1}{2}sqrt{6frac{3}{7}}

$$\frac { 4 } { 3 } \sqrt { 4 \frac { 2 } { 7 } } \times ( - \frac { 3 } { 2 } \sqrt { 2 \frac { 2 } { 3 } } ) \div \frac { 1 } { 2 } \sqrt { 6 \frac { 3 } { 7 } }$$

Evaluate

$$-\frac{240}{7}\approx -34.285714286$$

Solution Steps

Multiply 4 and 7 to get 28.

$$\frac{\frac{4}{3}\sqrt{\frac{28+2}{7}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Add 28 and 2 to get 30.

$$\frac{\frac{4}{3}\sqrt{\frac{30}{7}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Rewrite the square root of the division \sqrt{\frac{30}{7}} as the division of square roots \frac{\sqrt{30}}{\sqrt{7}}.

$$\frac{\frac{4}{3}\times \frac{\sqrt{30}}{\sqrt{7}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Rationalize the denominator of \frac{\sqrt{30}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\frac{4}{3}\times \frac{\sqrt{30}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{4}{3}\times \frac{\sqrt{30}\sqrt{7}}{7}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

To multiply \sqrt{30} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\frac{4}{3}\times \frac{\sqrt{210}}{7}\left(-\frac{3}{2}\right)\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply \frac{4}{3} times -\frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{4\left(-3\right)}{3\times 2}\times \frac{\sqrt{210}}{7}\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Do the multiplications in the fraction \frac{4\left(-3\right)}{3\times 2}.

$$\frac{\frac{-12}{6}\times \frac{\sqrt{210}}{7}\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Divide -12 by 6 to get -2.

$$\frac{-2\times \frac{\sqrt{210}}{7}\sqrt{\frac{2\times 3+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply 2 and 3 to get 6.

$$\frac{-2\times \frac{\sqrt{210}}{7}\sqrt{\frac{6+2}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Add 6 and 2 to get 8.

$$\frac{-2\times \frac{\sqrt{210}}{7}\sqrt{\frac{8}{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

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

$$\frac{-2\times \frac{\sqrt{210}}{7}\times \frac{\sqrt{8}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

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}.

$$\frac{-2\times \frac{\sqrt{210}}{7}\times \frac{2\sqrt{2}}{\sqrt{3}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

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

$$\frac{-2\times \frac{\sqrt{210}}{7}\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

The square of \sqrt{3} is 3.

$$\frac{-2\times \frac{\sqrt{210}}{7}\times \frac{2\sqrt{2}\sqrt{3}}{3}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

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

$$\frac{-2\times \frac{\sqrt{210}}{7}\times \frac{2\sqrt{6}}{3}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Express -2\times \frac{\sqrt{210}}{7} as a single fraction.

$$\frac{\frac{-2\sqrt{210}}{7}\times \frac{2\sqrt{6}}{3}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply \frac{-2\sqrt{210}}{7} times \frac{2\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{-2\sqrt{210}\times 2\sqrt{6}}{7\times 3}}{\frac{1}{2}}\sqrt{\frac{6\times 7+3}{7}}$$

Divide \frac{-2\sqrt{210}\times 2\sqrt{6}}{7\times 3} by \frac{1}{2} by multiplying \frac{-2\sqrt{210}\times 2\sqrt{6}}{7\times 3} by the reciprocal of \frac{1}{2}.

$$\frac{-2\sqrt{210}\times 2\sqrt{6}\times 2}{7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Cancel out -1 in both numerator and denominator.

$$\frac{2\sqrt{210}\times 2\sqrt{6}\times 2}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Factor 210=6\times 35. Rewrite the square root of the product \sqrt{6\times 35} as the product of square roots \sqrt{6}\sqrt{35}.

$$\frac{2\sqrt{6}\sqrt{35}\times 2\sqrt{6}\times 2}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{2\times 6\times 2\sqrt{35}\times 2}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply 2 and 6 to get 12.

$$\frac{12\times 2\sqrt{35}\times 2}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply 12 and 2 to get 24.

$$\frac{24\sqrt{35}\times 2}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply 24 and 2 to get 48.

$$\frac{48\sqrt{35}}{-7\times 3}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply -7 and 3 to get -21.

$$\frac{48\sqrt{35}}{-21}\sqrt{\frac{6\times 7+3}{7}}$$

Divide 48\sqrt{35} by -21 to get -\frac{16}{7}\sqrt{35}.

$$-\frac{16}{7}\sqrt{35}\sqrt{\frac{6\times 7+3}{7}}$$

Multiply 6 and 7 to get 42.

$$-\frac{16}{7}\sqrt{35}\sqrt{\frac{42+3}{7}}$$

Add 42 and 3 to get 45.

$$-\frac{16}{7}\sqrt{35}\sqrt{\frac{45}{7}}$$

Rewrite the square root of the division \sqrt{\frac{45}{7}} as the division of square roots \frac{\sqrt{45}}{\sqrt{7}}.

$$-\frac{16}{7}\sqrt{35}\times \frac{\sqrt{45}}{\sqrt{7}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$-\frac{16}{7}\sqrt{35}\times \frac{3\sqrt{5}}{\sqrt{7}}$$

Rationalize the denominator of \frac{3\sqrt{5}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$-\frac{16}{7}\sqrt{35}\times \frac{3\sqrt{5}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$-\frac{16}{7}\sqrt{35}\times \frac{3\sqrt{5}\sqrt{7}}{7}$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$-\frac{16}{7}\sqrt{35}\times \frac{3\sqrt{35}}{7}$$

Multiply -\frac{16}{7} times \frac{3\sqrt{35}}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-16\times 3\sqrt{35}}{7\times 7}\sqrt{35}$$

Multiply -16 and 3 to get -48.

$$\frac{-48\sqrt{35}}{7\times 7}\sqrt{35}$$

Multiply 7 and 7 to get 49.

$$\frac{-48\sqrt{35}}{49}\sqrt{35}$$

Express \frac{-48\sqrt{35}}{49}\sqrt{35} as a single fraction.

$$\frac{-48\sqrt{35}\sqrt{35}}{49}$$

Multiply \sqrt{35} and \sqrt{35} to get 35.

$$\frac{-48\times 35}{49}$$

Multiply -48 and 35 to get -1680.

$$\frac{-1680}{49}$$

Reduce the fraction \frac{-1680}{49} to lowest terms by extracting and canceling out 7.

$$-\frac{240}{7}$$

Factor

$$-\frac{240}{7} = -34\frac{2}{7} = -34.285714285714285$$