Solve sqrt{18}div(2sqrt{8}*1/3sqrt{54})

$$\sqrt { 18 } \div ( 2 \sqrt { 8 } \times \frac { 1 } { 3 } \sqrt { 54 } )$$

Evaluate

$$\frac{\sqrt{6}}{8}\approx 0.306186218$$

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}.

$$\frac{3\sqrt{2}}{2\sqrt{8}\times \frac{1}{3}\sqrt{54}}$$

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}.

$$\frac{3\sqrt{2}}{2\times 2\sqrt{2}\times \frac{1}{3}\sqrt{54}}$$

Multiply 2 and 2 to get 4.

$$\frac{3\sqrt{2}}{4\sqrt{2}\times \frac{1}{3}\sqrt{54}}$$

Multiply 4 and \frac{1}{3} to get \frac{4}{3}.

$$\frac{3\sqrt{2}}{\frac{4}{3}\sqrt{2}\sqrt{54}}$$

Factor 54=3^{2}\times 6. Rewrite the square root of the product \sqrt{3^{2}\times 6} as the product of square roots \sqrt{3^{2}}\sqrt{6}. Take the square root of 3^{2}.

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

Cancel out 3 and 3.

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

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}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Multiply 4 and 2 to get 8.

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

Multiply 8 and 3 to get 24.

$$\frac{3\sqrt{6}}{24}$$

Divide 3\sqrt{6} by 24 to get \frac{1}{8}\sqrt{6}.

$$\frac{1}{8}\sqrt{6}$$