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

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

Evaluate

$$\frac{3\left(\sqrt{3}+1\right)}{2}\approx 4.098076211$$

Short Solution Steps

Divide \sqrt{3} by \sqrt{3} to get 1.

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

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

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

The square of \sqrt{3} is 3.

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

Express 1\times \frac{\sqrt{3}}{3} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

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

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

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

Divide \sqrt{3} by \frac{3-\sqrt{3}}{3} by multiplying \sqrt{3} by the reciprocal of \frac{3-\sqrt{3}}{3}.

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

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

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

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

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

Square 3. Square \sqrt{3}.

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

Subtract 3 from 9 to get 6.

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

Express \frac{\sqrt{3}\times 3\left(3+\sqrt{3}\right)}{6}\times 1 as a single fraction.

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

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

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

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{3}+3\left(\sqrt{3}\right)^{2}}{6}$$

The square of \sqrt{3} is 3.

$$\frac{9\sqrt{3}+3\times 3}{6}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{3}+9}{6}$$

Factor

$$\frac{3 {(\sqrt{3} + 1)}}{2} = 4.098076211353316$$