Solve 3sqrt{3}xsqrt{3/8}div2sqrt{3}

$$3 \sqrt { 3 } x \sqrt { \frac { 3 } { 8 } } \div 2 \sqrt { 3 }$$

Evaluate

$$\frac{9\sqrt{6}x}{8}$$

Short Solution Steps

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

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply 2 and 2 to get 4.

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{6}x}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{9\sqrt{6}x}{8}$$

Differentiate w.r.t. x

$$\frac{9 \sqrt{6}}{8} = 2.7556759606310752$$