Solve (sqrt{3/4}+sqrt{4/3}):sqrt{12}

$$( \sqrt { \frac { 3 } { 4 } } + \sqrt { \frac { 4 } { 3 } } ) : \sqrt { 12 }$$

Evaluate

$$\frac{7}{12}\approx 0.583333333$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

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

Calculate the square root of 4 and get 2.

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

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

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

Calculate the square root of 4 and get 2.

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

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

$$\frac{\frac{\sqrt{3}}{2}+\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{12}}$$

The square of \sqrt{3} is 3.

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

Combine \frac{\sqrt{3}}{2} and \frac{2\sqrt{3}}{3} to get \frac{7}{6}\sqrt{3}.

$$\frac{\frac{7}{6}\sqrt{3}}{\sqrt{12}}$$

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}.

$$\frac{\frac{7}{6}\sqrt{3}}{2\sqrt{3}}$$

Cancel out \sqrt{3} in both numerator and denominator.

$$\frac{\frac{7}{6}}{2}$$

Express \frac{\frac{7}{6}}{2} as a single fraction.

$$\frac{7}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$\frac{7}{12}$$

Factor

$$\frac{7}{2 ^ {2} \cdot 3} = 0.5833333333333334$$