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

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

Evaluate

$$\frac{\sqrt{3}-1}{2}\approx 0.366025404$$

Solution Steps

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

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

Consider \left(6+2\sqrt{3}\right)\left(6-2\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate 6 to the power of 2 and get 36.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{36-\left(2\sqrt{3}\right)^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{36-2^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{36-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{36-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{36-12}$$

Subtract 12 from 36 to get 24.

$$\frac{2\sqrt{3}\left(6-2\sqrt{3}\right)}{24}$$

Divide 2\sqrt{3}\left(6-2\sqrt{3}\right) by 24 to get \frac{1}{12}\sqrt{3}\left(6-2\sqrt{3}\right).

$$\frac{1}{12}\sqrt{3}\left(6-2\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{12}\sqrt{3} by 6-2\sqrt{3}.

$$\frac{1}{12}\sqrt{3}\times 6+\frac{1}{12}\sqrt{3}\left(-2\right)\sqrt{3}$$

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

$$\frac{1}{12}\sqrt{3}\times 6+\frac{1}{12}\times 3\left(-2\right)$$

Multiply \frac{1}{12} and 6 to get \frac{6}{12}.

$$\frac{6}{12}\sqrt{3}+\frac{1}{12}\times 3\left(-2\right)$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

$$\frac{1}{2}\sqrt{3}+\frac{1}{12}\times 3\left(-2\right)$$

Multiply \frac{1}{12} and 3 to get \frac{3}{12}.

$$\frac{1}{2}\sqrt{3}+\frac{3}{12}\left(-2\right)$$

Reduce the fraction \frac{3}{12} to lowest terms by extracting and canceling out 3.

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

Multiply \frac{1}{4} and -2 to get \frac{-2}{4}.

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

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}\sqrt{3}-\frac{1}{2}$$