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

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

Evaluate

$$-\frac{4\sqrt{3}}{3}+1\approx -1.309401077$$

Solution Steps

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

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

Consider \left(2+\sqrt{3}\right)\left(2-\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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

The square of \sqrt{3} is 3.

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

Combine -\sqrt{3} and -\frac{\sqrt{3}}{3} to get -\frac{4}{3}\sqrt{3}.

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

Subtract 2 from 3 to get 1.

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

Calculate the square root of 1 and get 1.

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

Anything divided by one gives itself.

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

Subtract 1 from 2 to get 1.

$$1-\frac{4}{3}\sqrt{3}$$