Solve (sqrt{24-3}sqrt{15+2}sqrt{22/3})*1/2

$$( \sqrt{ 24-3 } \sqrt{ 15+2 } \sqrt{ 2 \frac{ 2 }{ 3 } } ) \times \frac{ 1 }{ 2 }$$

Evaluate

$$\sqrt{238}\approx 15.427248621$$

Solution Steps

Subtract 3 from 24 to get 21.

$$\sqrt{21}\sqrt{15+2}\sqrt{\frac{2\times 3+2}{3}}\times \frac{1}{2}$$

Add 15 and 2 to get 17.

$$\sqrt{21}\sqrt{17}\sqrt{\frac{2\times 3+2}{3}}\times \frac{1}{2}$$

Multiply 2 and 3 to get 6.

$$\sqrt{21}\sqrt{17}\sqrt{\frac{6+2}{3}}\times \frac{1}{2}$$

Add 6 and 2 to get 8.

$$\sqrt{21}\sqrt{17}\sqrt{\frac{8}{3}}\times \frac{1}{2}$$

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

$$\sqrt{21}\sqrt{17}\times \frac{\sqrt{8}}{\sqrt{3}}\times \frac{1}{2}$$

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}.

$$\sqrt{21}\sqrt{17}\times \frac{2\sqrt{2}}{\sqrt{3}}\times \frac{1}{2}$$

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

$$\sqrt{21}\sqrt{17}\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times \frac{1}{2}$$

The square of \sqrt{3} is 3.

$$\sqrt{21}\sqrt{17}\times \frac{2\sqrt{2}\sqrt{3}}{3}\times \frac{1}{2}$$

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

$$\sqrt{21}\sqrt{17}\times \frac{2\sqrt{6}}{3}\times \frac{1}{2}$$

To multiply \sqrt{21} and \sqrt{17}, multiply the numbers under the square root.

$$\sqrt{357}\times \frac{2\sqrt{6}}{3}\times \frac{1}{2}$$

Express \sqrt{21}\times \frac{2\sqrt{6}}{3} as a single fraction.

$$\frac{\sqrt{21}\times 2\sqrt{6}}{3}\times \frac{1}{2}$$

Multiply \frac{\sqrt{21}\times 2\sqrt{6}}{3} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{21}\times 2\sqrt{6}}{3\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{6}\sqrt{21}}{3}$$

To multiply \sqrt{6} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{\sqrt{126}}{3}$$

Factor 126=3^{2}\times 14. Rewrite the square root of the product \sqrt{3^{2}\times 14} as the product of square roots \sqrt{3^{2}}\sqrt{14}. Take the square root of 3^{2}.

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

Cancel out 3 and 3.

$$\sqrt{14}$$