Solve sqrt{14}divsqrt{6}*sqrt{27/2}

$$\sqrt { 14 } \div \sqrt { 6 } \times \sqrt { \frac { 27 } { 2 } }$$

Evaluate

$$\frac{3\sqrt{14}}{2}\approx 5.61248608$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{14}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\sqrt{\frac{27}{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{14}\sqrt{6}}{6}\sqrt{\frac{27}{2}}$$

To multiply \sqrt{14} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{84}}{6}\sqrt{\frac{27}{2}}$$

Factor 84=2^{2}\times 21. Rewrite the square root of the product \sqrt{2^{2}\times 21} as the product of square roots \sqrt{2^{2}}\sqrt{21}. Take the square root of 2^{2}.

$$\frac{2\sqrt{21}}{6}\sqrt{\frac{27}{2}}$$

Divide 2\sqrt{21} by 6 to get \frac{1}{3}\sqrt{21}.

$$\frac{1}{3}\sqrt{21}\sqrt{\frac{27}{2}}$$

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

$$\frac{1}{3}\sqrt{21}\times \frac{\sqrt{27}}{\sqrt{2}}$$

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}.

$$\frac{1}{3}\sqrt{21}\times \frac{3\sqrt{3}}{\sqrt{2}}$$

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

$$\frac{1}{3}\sqrt{21}\times \frac{3\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{1}{3}\sqrt{21}\times \frac{3\sqrt{3}\sqrt{2}}{2}$$

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

$$\frac{1}{3}\sqrt{21}\times \frac{3\sqrt{6}}{2}$$

Multiply \frac{1}{3} times \frac{3\sqrt{6}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{6}}{3\times 2}\sqrt{21}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\sqrt{6}}{2}\sqrt{21}$$

Express \frac{\sqrt{6}}{2}\sqrt{21} as a single fraction.

$$\frac{\sqrt{6}\sqrt{21}}{2}$$

To multiply \sqrt{6} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{\sqrt{126}}{2}$$

Factor 126=3^{2}\times 14. Rewrite the square root of the product \sqrt{3^{2}\times 14} as the product of square roots \sqrt{3^{2}}\sqrt{14}. Take the square root of 3^{2}.

$$\frac{3\sqrt{14}}{2}$$