Solve frac{2sqrt{3}*3sqrt{5}*2sqrt{6}}{sqrt{18}}

$$\frac { 2 \sqrt { 3 } \cdot 3 \sqrt { 5 } \cdot 2 \sqrt { 6 } } { \sqrt { 18 } }$$

Evaluate

$$12\sqrt{5}\approx 26.83281573$$

Solution Steps

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}.

$$\frac{2\times 2\times 3\sqrt{3}\sqrt{5}\sqrt{3}\sqrt{2}}{\sqrt{18}}$$

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

$$\frac{2\times 2\times 3\times 3\sqrt{5}\sqrt{2}}{\sqrt{18}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 3\times 3\sqrt{5}\sqrt{2}}{\sqrt{18}}$$

Multiply 4 and 3 to get 12.

$$\frac{12\times 3\sqrt{5}\sqrt{2}}{\sqrt{18}}$$

Multiply 12 and 3 to get 36.

$$\frac{36\sqrt{5}\sqrt{2}}{\sqrt{18}}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{36\sqrt{10}}{\sqrt{18}}$$

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}.

$$\frac{36\sqrt{10}}{3\sqrt{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{12\sqrt{10}}{\sqrt{2}}$$

Rationalize the denominator of \frac{12\sqrt{10}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{12\sqrt{10}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{12\sqrt{10}\sqrt{2}}{2}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{12\sqrt{2}\sqrt{5}\sqrt{2}}{2}$$

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

$$\frac{12\times 2\sqrt{5}}{2}$$

Cancel out 2 and 2.

$$12\sqrt{5}$$