Solve 1/2sqrt{3-1}+3/sqrt{3+1}

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

Evaluate

$$\frac{37\sqrt{3}-31}{22}\approx 1.503903631$$

Short Solution Steps

Rationalize the denominator of \frac{1}{2\sqrt{3}-1} by multiplying numerator and denominator by 2\sqrt{3}+1.

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

Consider \left(2\sqrt{3}-1\right)\left(2\sqrt{3}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{3}+1}{4\left(\sqrt{3}\right)^{2}-1^{2}}+\frac{3}{\sqrt{3}+1}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{3}+1}{4\times 3-1^{2}}+\frac{3}{\sqrt{3}+1}$$

Multiply 4 and 3 to get 12.

$$\frac{2\sqrt{3}+1}{12-1^{2}}+\frac{3}{\sqrt{3}+1}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{2\sqrt{3}+1}{12-1}+\frac{3}{\sqrt{3}+1}$$

Subtract 1 from 12 to get 11.

$$\frac{2\sqrt{3}+1}{11}+\frac{3}{\sqrt{3}+1}$$

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

$$\frac{2\sqrt{3}+1}{11}+\frac{3\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}$$

Consider \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{3}+1}{11}+\frac{3\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}$$

Square \sqrt{3}. Square 1.

$$\frac{2\sqrt{3}+1}{11}+\frac{3\left(\sqrt{3}-1\right)}{3-1}$$

Subtract 1 from 3 to get 2.

$$\frac{2\sqrt{3}+1}{11}+\frac{3\left(\sqrt{3}-1\right)}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 11 and 2 is 22. Multiply \frac{2\sqrt{3}+1}{11} times \frac{2}{2}. Multiply \frac{3\left(\sqrt{3}-1\right)}{2} times \frac{11}{11}.

$$\frac{2\left(2\sqrt{3}+1\right)}{22}+\frac{11\times 3\left(\sqrt{3}-1\right)}{22}$$

Since \frac{2\left(2\sqrt{3}+1\right)}{22} and \frac{11\times 3\left(\sqrt{3}-1\right)}{22} have the same denominator, add them by adding their numerators.

$$\frac{2\left(2\sqrt{3}+1\right)+11\times 3\left(\sqrt{3}-1\right)}{22}$$

Do the multiplications in 2\left(2\sqrt{3}+1\right)+11\times 3\left(\sqrt{3}-1\right).

$$\frac{4\sqrt{3}+2+33\sqrt{3}-33}{22}$$

Do the calculations in 4\sqrt{3}+2+33\sqrt{3}-33.

$$\frac{37\sqrt{3}-31}{22}$$