Solve sqrt{4/3}divsqrt{3/2}*sqrt{9/8}

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

Evaluate

$$1$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

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

Calculate the square root of 4 and get 2.

$$\frac{\frac{2}{\sqrt{3}}}{\sqrt{\frac{3}{2}}}\sqrt{\frac{9}{8}}$$

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

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

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{3}}{3}}{\sqrt{\frac{3}{2}}}\sqrt{\frac{9}{8}}$$

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{\sqrt{3}}{\sqrt{2}}}\sqrt{\frac{9}{8}}$$

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

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

The square of \sqrt{2} is 2.

$$\frac{\frac{2\sqrt{3}}{3}}{\frac{\sqrt{3}\sqrt{2}}{2}}\sqrt{\frac{9}{8}}$$

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

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

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

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

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

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

The square of \sqrt{6} is 6.

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

Multiply 2 and 2 to get 4.

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

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}.

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

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

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

Multiply 3 and 6 to get 18.

$$\frac{4\times 3\sqrt{2}}{18}\sqrt{\frac{9}{8}}$$

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{2}}{18}\sqrt{\frac{9}{8}}$$

Divide 12\sqrt{2} by 18 to get \frac{2}{3}\sqrt{2}.

$$\frac{2}{3}\sqrt{2}\sqrt{\frac{9}{8}}$$

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

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

Calculate the square root of 9 and get 3.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

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

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

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

$$\frac{\sqrt{2}}{2}\sqrt{2}$$

Express \frac{\sqrt{2}}{2}\sqrt{2} as a single fraction.

$$\frac{\sqrt{2}\sqrt{2}}{2}$$

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

$$\frac{2}{2}$$

Divide 2 by 2 to get 1.

$$1$$

Factor

$$1$$