Solve 3/36-18sqrt{3}

$$\frac { 3 } { 36 - 18 \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{3}+2}{6}\approx 0.622008468$$

Solution Steps

Rationalize the denominator of \frac{3}{36-18\sqrt{3}} by multiplying numerator and denominator by 36+18\sqrt{3}.

$$\frac{3\left(36+18\sqrt{3}\right)}{\left(36-18\sqrt{3}\right)\left(36+18\sqrt{3}\right)}$$

Consider \left(36-18\sqrt{3}\right)\left(36+18\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}.

$$\frac{3\left(36+18\sqrt{3}\right)}{36^{2}-\left(-18\sqrt{3}\right)^{2}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-\left(-18\sqrt{3}\right)^{2}}$$

Expand \left(-18\sqrt{3}\right)^{2}.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-\left(-18\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -18 to the power of 2 and get 324.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-324\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-324\times 3}$$

Multiply 324 and 3 to get 972.

$$\frac{3\left(36+18\sqrt{3}\right)}{1296-972}$$

Subtract 972 from 1296 to get 324.

$$\frac{3\left(36+18\sqrt{3}\right)}{324}$$

Divide 3\left(36+18\sqrt{3}\right) by 324 to get \frac{1}{108}\left(36+18\sqrt{3}\right).

$$\frac{1}{108}\left(36+18\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{108} by 36+18\sqrt{3}.

$$\frac{1}{108}\times 36+\frac{1}{108}\times 18\sqrt{3}$$

Multiply \frac{1}{108} and 36 to get \frac{36}{108}.

$$\frac{36}{108}+\frac{1}{108}\times 18\sqrt{3}$$

Reduce the fraction \frac{36}{108} to lowest terms by extracting and canceling out 36.

$$\frac{1}{3}+\frac{1}{108}\times 18\sqrt{3}$$

Multiply \frac{1}{108} and 18 to get \frac{18}{108}.

$$\frac{1}{3}+\frac{18}{108}\sqrt{3}$$

Reduce the fraction \frac{18}{108} to lowest terms by extracting and canceling out 18.

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