Solve frac{1/3+sqrt{3}}{1-1/3*sqrt{3}}

$$\frac{ \frac{ 1 }{ 3 } + \sqrt{ 3 } }{ 1- \frac{ 1 }{ 3 } \times \sqrt{ 3 } }$$

Evaluate

$$\frac{5\sqrt{3}}{3}+2\approx 4.886751346$$

Solution Steps

Rationalize the denominator of \frac{\frac{1}{3}+\sqrt{3}}{1-\frac{1}{3}\sqrt{3}} by multiplying numerator and denominator by 1+\frac{1}{3}\sqrt{3}.

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

Consider \left(1-\frac{1}{3}\sqrt{3}\right)\left(1+\frac{1}{3}\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}.

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

Calculate 1 to the power of 2 and get 1.

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

Expand \left(-\frac{1}{3}\sqrt{3}\right)^{2}.

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

Calculate -\frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{\left(\frac{1}{3}+\sqrt{3}\right)\left(1+\frac{1}{3}\sqrt{3}\right)}{1-\frac{1}{9}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(\frac{1}{3}+\sqrt{3}\right)\left(1+\frac{1}{3}\sqrt{3}\right)}{1-\frac{1}{9}\times 3}$$

Multiply \frac{1}{9} and 3 to get \frac{3}{9}.

$$\frac{\left(\frac{1}{3}+\sqrt{3}\right)\left(1+\frac{1}{3}\sqrt{3}\right)}{1-\frac{3}{9}}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

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

Convert 1 to fraction \frac{3}{3}.

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

Since \frac{3}{3} and \frac{1}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract 1 from 3 to get 2.

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

Divide \left(\frac{1}{3}+\sqrt{3}\right)\left(1+\frac{1}{3}\sqrt{3}\right) by \frac{2}{3} by multiplying \left(\frac{1}{3}+\sqrt{3}\right)\left(1+\frac{1}{3}\sqrt{3}\right) by the reciprocal of \frac{2}{3}.

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

Apply the distributive property by multiplying each term of \frac{1}{3}+\sqrt{3} by each term of 1+\frac{1}{3}\sqrt{3}.

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

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

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

Multiply \frac{1}{3} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 1}{3\times 3}.

$$\frac{\left(\frac{1}{3}+\frac{1}{9}\sqrt{3}+\sqrt{3}+3\times \frac{1}{3}\right)\times 3}{2}$$

Combine \frac{1}{9}\sqrt{3} and \sqrt{3} to get \frac{10}{9}\sqrt{3}.

$$\frac{\left(\frac{1}{3}+\frac{10}{9}\sqrt{3}+3\times \frac{1}{3}\right)\times 3}{2}$$

Cancel out 3 and 3.

$$\frac{\left(\frac{1}{3}+\frac{10}{9}\sqrt{3}+1\right)\times 3}{2}$$

Convert 1 to fraction \frac{3}{3}.

$$\frac{\left(\frac{1}{3}+\frac{10}{9}\sqrt{3}+\frac{3}{3}\right)\times 3}{2}$$

Since \frac{1}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

$$\frac{\left(\frac{1+3}{3}+\frac{10}{9}\sqrt{3}\right)\times 3}{2}$$

Add 1 and 3 to get 4.

$$\frac{\left(\frac{4}{3}+\frac{10}{9}\sqrt{3}\right)\times 3}{2}$$

Use the distributive property to multiply \frac{4}{3}+\frac{10}{9}\sqrt{3} by 3.

$$\frac{\frac{4}{3}\times 3+\frac{10}{9}\sqrt{3}\times 3}{2}$$

Cancel out 3 and 3.

$$\frac{4+\frac{10}{9}\sqrt{3}\times 3}{2}$$

Express \frac{10}{9}\times 3 as a single fraction.

$$\frac{4+\frac{10\times 3}{9}\sqrt{3}}{2}$$

Multiply 10 and 3 to get 30.

$$\frac{4+\frac{30}{9}\sqrt{3}}{2}$$

Reduce the fraction \frac{30}{9} to lowest terms by extracting and canceling out 3.

$$\frac{4+\frac{10}{3}\sqrt{3}}{2}$$

Factor

$$\frac{5 \sqrt{3} + 6}{3} = 4.886751345948128$$