Solve (sqrt{18}+1/4sqrt{6})divsqrt{12}

$$( \sqrt { 18 } + \frac { 1 } { 4 } \sqrt { 6 } ) \div \sqrt { 12 }$$

Evaluate

$$\frac{\sqrt{2}}{8}+\frac{\sqrt{6}}{2}\approx 1.401521567$$

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}.

$$\frac{3\sqrt{2}+\frac{1}{4}\sqrt{6}}{\sqrt{12}}$$

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}.

$$\frac{3\sqrt{2}+\frac{1}{4}\sqrt{6}}{2\sqrt{3}}$$

Rationalize the denominator of \frac{3\sqrt{2}+\frac{1}{4}\sqrt{6}}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(3\sqrt{2}+\frac{1}{4}\sqrt{6}\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3\sqrt{2}+\frac{1}{4}\sqrt{6}\right)\sqrt{3}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{\left(3\sqrt{2}+\frac{1}{4}\sqrt{6}\right)\sqrt{3}}{6}$$

Use the distributive property to multiply 3\sqrt{2}+\frac{1}{4}\sqrt{6} by \sqrt{3}.

$$\frac{3\sqrt{2}\sqrt{3}+\frac{1}{4}\sqrt{6}\sqrt{3}}{6}$$

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

$$\frac{3\sqrt{6}+\frac{1}{4}\sqrt{6}\sqrt{3}}{6}$$

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}.

$$\frac{3\sqrt{6}+\frac{1}{4}\sqrt{3}\sqrt{2}\sqrt{3}}{6}$$

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

$$\frac{3\sqrt{6}+\frac{1}{4}\times 3\sqrt{2}}{6}$$

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

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

Factor

$$\frac{\sqrt{2} + 4 \sqrt{6}}{8} = 1.401521566688226$$