Solve frac{4sqrt{3}+3sqrt{6}}{2sqrt{3}}

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

Evaluate

$$\frac{3\sqrt{2}}{2}+2\approx 4.121320344$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

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}.

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

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

$$\frac{12+3\times 3\sqrt{2}}{6}$$

Multiply 3 and 3 to get 9.

$$\frac{12+9\sqrt{2}}{6}$$

Factor

$$\frac{3 \sqrt{2} + 4}{2} = 4.121320343559643$$