Solve sqrt{12}-(1/1-sqrt{3})

$$\sqrt { 12 } - ( \frac { 1 } { 1 - \sqrt { 3 } } )$$

Evaluate

$$\frac{5\sqrt{3}+1}{2}\approx 4.830127019$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

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

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Multiply both numerator and denominator by -1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3} times \frac{2}{2}.

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

Since \frac{2\times 2\sqrt{3}}{2} and \frac{-1-\sqrt{3}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 2\sqrt{3}-\left(-1-\sqrt{3}\right).

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

Do the calculations in 4\sqrt{3}+1+\sqrt{3}.

$$\frac{5\sqrt{3}+1}{2}$$