Solve 1/12+13/3/1-frac{11{12}*frac{sqrt{3}}{3}}

$$\frac { \frac { 1 } { 12 } + \frac { 13 } { 3 } } { 1 - \frac { 11 } { 12 } \times \frac { \sqrt { 3 } } { 3 } }$$

Evaluate

$$\frac{583\sqrt{3}+1908}{311}\approx 9.381947334$$

Short Solution Steps

Least common multiple of 12 and 3 is 12. Convert \frac{1}{12} and \frac{13}{3} to fractions with denominator 12.

$$\frac{\frac{1}{12}+\frac{52}{12}}{1-\frac{11}{12}\times \frac{\sqrt{3}}{3}}$$

Since \frac{1}{12} and \frac{52}{12} have the same denominator, add them by adding their numerators.

$$\frac{\frac{1+52}{12}}{1-\frac{11}{12}\times \frac{\sqrt{3}}{3}}$$

Add 1 and 52 to get 53.

$$\frac{\frac{53}{12}}{1-\frac{11}{12}\times \frac{\sqrt{3}}{3}}$$

Multiply \frac{11}{12} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{53}{12}}{1-\frac{11\sqrt{3}}{12\times 3}}$$

Multiply 12 and 3 to get 36.

$$\frac{\frac{53}{12}}{1-\frac{11\sqrt{3}}{36}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{36}{36}.

$$\frac{\frac{53}{12}}{\frac{36}{36}-\frac{11\sqrt{3}}{36}}$$

Since \frac{36}{36} and \frac{11\sqrt{3}}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{53}{12}}{\frac{36-11\sqrt{3}}{36}}$$

Divide \frac{53}{12} by \frac{36-11\sqrt{3}}{36} by multiplying \frac{53}{12} by the reciprocal of \frac{36-11\sqrt{3}}{36}.

$$\frac{53\times 36}{12\left(36-11\sqrt{3}\right)}$$

Cancel out 12 in both numerator and denominator.

$$\frac{3\times 53}{-11\sqrt{3}+36}$$

Rationalize the denominator of \frac{3\times 53}{-11\sqrt{3}+36} by multiplying numerator and denominator by -11\sqrt{3}-36.

$$\frac{3\times 53\left(-11\sqrt{3}-36\right)}{\left(-11\sqrt{3}+36\right)\left(-11\sqrt{3}-36\right)}$$

Consider \left(-11\sqrt{3}+36\right)\left(-11\sqrt{3}-36\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\times 53\left(-11\sqrt{3}-36\right)}{\left(-11\sqrt{3}\right)^{2}-36^{2}}$$

Multiply 3 and 53 to get 159.

$$\frac{159\left(-11\sqrt{3}-36\right)}{\left(-11\sqrt{3}\right)^{2}-36^{2}}$$

Expand \left(-11\sqrt{3}\right)^{2}.

$$\frac{159\left(-11\sqrt{3}-36\right)}{\left(-11\right)^{2}\left(\sqrt{3}\right)^{2}-36^{2}}$$

Calculate -11 to the power of 2 and get 121.

$$\frac{159\left(-11\sqrt{3}-36\right)}{121\left(\sqrt{3}\right)^{2}-36^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{159\left(-11\sqrt{3}-36\right)}{121\times 3-36^{2}}$$

Multiply 121 and 3 to get 363.

$$\frac{159\left(-11\sqrt{3}-36\right)}{363-36^{2}}$$

Calculate 36 to the power of 2 and get 1296.

$$\frac{159\left(-11\sqrt{3}-36\right)}{363-1296}$$

Subtract 1296 from 363 to get -933.

$$\frac{159\left(-11\sqrt{3}-36\right)}{-933}$$

Divide 159\left(-11\sqrt{3}-36\right) by -933 to get -\frac{53}{311}\left(-11\sqrt{3}-36\right).

$$-\frac{53}{311}\left(-11\sqrt{3}-36\right)$$

Use the distributive property to multiply -\frac{53}{311} by -11\sqrt{3}-36.

$$-\frac{53}{311}\left(-11\right)\sqrt{3}-\frac{53}{311}\left(-36\right)$$

Express -\frac{53}{311}\left(-11\right) as a single fraction.

$$\frac{-53\left(-11\right)}{311}\sqrt{3}-\frac{53}{311}\left(-36\right)$$

Multiply -53 and -11 to get 583.

$$\frac{583}{311}\sqrt{3}-\frac{53}{311}\left(-36\right)$$

Express -\frac{53}{311}\left(-36\right) as a single fraction.

$$\frac{583}{311}\sqrt{3}+\frac{-53\left(-36\right)}{311}$$

Multiply -53 and -36 to get 1908.

$$\frac{583}{311}\sqrt{3}+\frac{1908}{311}$$

Factor

$$\frac{53 {(11 \sqrt{3} + 36)}}{311} = 9.381947333802751$$