Solve frac{(4sqrt{3}+1/4sqrt{6})}{sqrt{27}}

$$\frac { ( 4 \sqrt { 3 } + \frac { 1 } { 4 } \sqrt { 6 } ) } { \sqrt { 27 } }$$

Evaluate

$$\frac{\sqrt{2}}{12}+\frac{4}{3}\approx 1.451184464$$

Solution Steps

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}.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

$$\frac{\left(4\sqrt{3}+\frac{1}{4}\sqrt{6}\right)\sqrt{3}}{9}$$

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

$$\frac{4\left(\sqrt{3}\right)^{2}+\frac{1}{4}\sqrt{6}\sqrt{3}}{9}$$

The square of \sqrt{3} is 3.

$$\frac{4\times 3+\frac{1}{4}\sqrt{6}\sqrt{3}}{9}$$

Multiply 4 and 3 to get 12.

$$\frac{12+\frac{1}{4}\sqrt{6}\sqrt{3}}{9}$$

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}.

$$\frac{12+\frac{1}{4}\sqrt{3}\sqrt{2}\sqrt{3}}{9}$$

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

$$\frac{12+\frac{1}{4}\times 3\sqrt{2}}{9}$$

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

$$\frac{12+\frac{3}{4}\sqrt{2}}{9}$$

Factor

$$\frac{\sqrt{2} + 16}{12} = 1.4511844635310913$$