Solve 2/sqrt{3}+12/sqrt{27}+2sqrt{7}

$$\frac { 2 } { \sqrt { 3 } } + \frac { 12 } { \sqrt { 27 } } + 2 \sqrt { 7 }$$

Evaluate

$$2\left(\sqrt{3}+\sqrt{7}\right)\approx 8.755604237$$

Solution Steps

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

$$\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\frac{12}{\sqrt{27}}+2\sqrt{7}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}+\frac{12}{\sqrt{27}}+2\sqrt{7}$$

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}.

$$\frac{2\sqrt{3}}{3}+\frac{12}{3\sqrt{3}}+2\sqrt{7}$$

Rationalize the denominator of \frac{12}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{3}}{3}+\frac{12\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}+2\sqrt{7}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}}{3}+\frac{12\sqrt{3}}{3\times 3}+2\sqrt{7}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\sqrt{3}}{3}+\frac{4\sqrt{3}}{3}+2\sqrt{7}$$

Combine \frac{2\sqrt{3}}{3} and \frac{4\sqrt{3}}{3} to get 2\sqrt{3}.

$$2\sqrt{3}+2\sqrt{7}$$

Factor

$$2 {(\sqrt{3} + \sqrt{7})} = 8.755604237$$