Solve frac{sqrt{72}}{sqrt{6}}quadtext{(2)}sqrt{18}divsqrt{8}

$$\frac { \sqrt { 72 } } { \sqrt { 6 } } \quad \text { (2) } \sqrt { 18 } \div \sqrt { 8 }$$

Evaluate

$$6\sqrt{3}\approx 10.392304845$$

Solution Steps

Rewrite the division of square roots \frac{\sqrt{72}}{\sqrt{6}} as the square root of the division \sqrt{\frac{72}{6}} and perform the division.

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

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{3}\sqrt{18}}{\sqrt{8}}$$

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}.

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

Multiply 4 and 3 to get 12.

$$\frac{12\sqrt{3}\sqrt{2}}{\sqrt{8}}$$

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

$$\frac{12\sqrt{6}}{\sqrt{8}}$$

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}.

$$\frac{12\sqrt{6}}{2\sqrt{2}}$$

Cancel out 2 in both numerator and denominator.

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

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

$$\frac{6\sqrt{6}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

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}.

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

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

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

Cancel out 2 and 2.

$$6\sqrt{3}$$