Solve sqrt{32/3}*(sqrt{27/2}divsqrt{9/8})*sqrt{2}

$$\sqrt { \frac { 32 } { 3 } } \cdot ( \sqrt { \frac { 27 } { 2 } } \div \sqrt { \frac { 9 } { 8 } } ) \cdot \sqrt { 2 }$$

Evaluate

$$16$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{32}{3}} as the division of square roots \frac{\sqrt{32}}{\sqrt{3}}.

$$\frac{\sqrt{32}}{\sqrt{3}}\times \frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{4\sqrt{2}}{\sqrt{3}}\times \frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

Rationalize the denominator of \frac{4\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{4\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

The square of \sqrt{3} is 3.

$$\frac{4\sqrt{2}\sqrt{3}}{3}\times \frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

Rewrite the square root of the division \sqrt{\frac{27}{2}} as the division of square roots \frac{\sqrt{27}}{\sqrt{2}}.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{\sqrt{27}}{\sqrt{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{3}}{\sqrt{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{3}\sqrt{2}}{2}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\sqrt{\frac{9}{8}}}\sqrt{2}$$

Rewrite the square root of the division \sqrt{\frac{9}{8}} as the division of square roots \frac{\sqrt{9}}{\sqrt{8}}.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{9}}{\sqrt{8}}}\sqrt{2}$$

Calculate the square root of 9 and get 3.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{3}{\sqrt{8}}}\sqrt{2}$$

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}.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{3}{2\sqrt{2}}}\sqrt{2}$$

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

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{3\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{3\sqrt{2}}{2\times 2}}\sqrt{2}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{6}}{3}\times \frac{\frac{3\sqrt{6}}{2}}{\frac{3\sqrt{2}}{4}}\sqrt{2}$$

Divide \frac{3\sqrt{6}}{2} by \frac{3\sqrt{2}}{4} by multiplying \frac{3\sqrt{6}}{2} by the reciprocal of \frac{3\sqrt{2}}{4}.

$$\frac{4\sqrt{6}}{3}\times \frac{3\sqrt{6}\times 4}{2\times 3\sqrt{2}}\sqrt{2}$$

Cancel out 2\times 3 in both numerator and denominator.

$$\frac{4\sqrt{6}}{3}\times \frac{2\sqrt{6}}{\sqrt{2}}\sqrt{2}$$

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{4\sqrt{6}}{3}\times \frac{2\sqrt{6}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{6}}{3}\times \frac{2\sqrt{6}\sqrt{2}}{2}\sqrt{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{4\sqrt{6}}{3}\times \frac{2\sqrt{2}\sqrt{3}\sqrt{2}}{2}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{4\sqrt{6}}{3}\times \frac{2\times 2\sqrt{3}}{2}\sqrt{2}$$

Cancel out 2 and 2.

$$\frac{4\sqrt{6}}{3}\times 2\sqrt{3}\sqrt{2}$$

Express \frac{4\sqrt{6}}{3}\times 2 as a single fraction.

$$\frac{4\sqrt{6}\times 2}{3}\sqrt{3}\sqrt{2}$$

Express \frac{4\sqrt{6}\times 2}{3}\sqrt{3} as a single fraction.

$$\frac{4\sqrt{6}\times 2\sqrt{3}}{3}\sqrt{2}$$

Express \frac{4\sqrt{6}\times 2\sqrt{3}}{3}\sqrt{2} as a single fraction.

$$\frac{4\sqrt{6}\times 2\sqrt{3}\sqrt{2}}{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{4\sqrt{3}\sqrt{2}\times 2\sqrt{3}\sqrt{2}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{4\times 3\times 2\sqrt{2}\sqrt{2}}{3}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{4\times 3\times 2\times 2}{3}$$

Multiply 4 and 3 to get 12.

$$\frac{12\times 2\times 2}{3}$$

Multiply 12 and 2 to get 24.

$$\frac{24\times 2}{3}$$

Multiply 24 and 2 to get 48.

$$\frac{48}{3}$$

Divide 48 by 3 to get 16.

$$16$$

Factor

$$2^{4}$$