Solve frac{sqrt{18}}{8}sqrt{27/2}

$$\frac{ \sqrt{ 18 } }{ 8 } \sqrt{ \frac{ 27 }{ 2 } }$$

Evaluate

$$\frac{9\sqrt{3}}{8}\approx 1.948557159$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{3\sqrt{2}}{8}\sqrt{\frac{27}{2}}$$

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

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply \frac{3\sqrt{2}}{8} times \frac{3\sqrt{6}}{2} by multiplying numerator times numerator and denominator times denominator.

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

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}.

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

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

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

Multiply 3 and 2 to get 6.

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

Multiply 6 and 3 to get 18.

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

Multiply 8 and 2 to get 16.

$$\frac{18\sqrt{3}}{16}$$

Divide 18\sqrt{3} by 16 to get \frac{9}{8}\sqrt{3}.

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