Solve frac{sqrt{30}3/2sqrt{22/3}}{-2sqrt{21/2}}

$$\frac{ \sqrt{ 30 } \frac{ 3 }{ 2 } \sqrt{ 2 \frac{ 2 }{ 3 } } }{ -2 \sqrt{ 2 \frac{ 1 }{ 2 } } }$$

Evaluate

$$-3\sqrt{2}\approx -4.242640687$$

Solution Steps

Multiply 2 and 3 to get 6.

$$\frac{\sqrt{30}\times \frac{3}{2}\sqrt{\frac{6+2}{3}}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

Add 6 and 2 to get 8.

$$\frac{\sqrt{30}\times \frac{3}{2}\sqrt{\frac{8}{3}}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

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

$$\frac{\sqrt{30}\times \frac{3}{2}\times \frac{\sqrt{8}}{\sqrt{3}}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

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}.

$$\frac{\sqrt{30}\times \frac{3}{2}\times \frac{2\sqrt{2}}{\sqrt{3}}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{\sqrt{30}\times \frac{3}{2}\times \frac{2\sqrt{2}\sqrt{3}}{3}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

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

$$\frac{\sqrt{30}\times \frac{3}{2}\times \frac{2\sqrt{6}}{3}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

Express \sqrt{30}\times \frac{2\sqrt{6}}{3} as a single fraction.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\sqrt{\frac{2\times 2+1}{2}}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\sqrt{\frac{4+1}{2}}}$$

Add 4 and 1 to get 5.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\sqrt{\frac{5}{2}}}$$

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

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\times \frac{\sqrt{5}}{\sqrt{2}}}$$

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

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\times \frac{\sqrt{5}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\times \frac{\sqrt{5}\sqrt{2}}{2}}$$

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

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-2\times \frac{\sqrt{10}}{2}}$$

Cancel out 2 and 2.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-\sqrt{10}}$$

Rationalize the denominator of \frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}}{-\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}\sqrt{10}}{-\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\frac{\sqrt{30}\times 2\sqrt{6}}{3}\times \frac{3}{2}\sqrt{10}}{-10}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$\frac{\frac{\sqrt{6}\sqrt{5}\times 2\sqrt{6}}{3}\times \frac{3}{2}\sqrt{10}}{-10}$$

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

$$\frac{\frac{6\times 2\sqrt{5}}{3}\times \frac{3}{2}\sqrt{10}}{-10}$$

Multiply 6 and 2 to get 12.

$$\frac{\frac{12\sqrt{5}}{3}\times \frac{3}{2}\sqrt{10}}{-10}$$

Divide 12\sqrt{5} by 3 to get 4\sqrt{5}.

$$\frac{4\sqrt{5}\times \frac{3}{2}\sqrt{10}}{-10}$$

Express 4\times \frac{3}{2} as a single fraction.

$$\frac{\frac{4\times 3}{2}\sqrt{5}\sqrt{10}}{-10}$$

Multiply 4 and 3 to get 12.

$$\frac{\frac{12}{2}\sqrt{5}\sqrt{10}}{-10}$$

Divide 12 by 2 to get 6.

$$\frac{6\sqrt{5}\sqrt{10}}{-10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{6\sqrt{5}\sqrt{5}\sqrt{2}}{-10}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{6\times 5\sqrt{2}}{-10}$$

Multiply 6 and 5 to get 30.

$$\frac{30\sqrt{2}}{-10}$$

Divide 30\sqrt{2} by -10 to get -3\sqrt{2}.

$$-3\sqrt{2}$$