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

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

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

Solution Steps

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

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

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

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

Square \sqrt{3}. Square 2.

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

Subtract 4 from 3 to get -1.

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

Anything divided by -1 gives its opposite. To find the opposite of \sqrt{3}-2, find the opposite of each term.

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

To find the opposite of -\sqrt{3}-\left(-2\right), find the opposite of each term.

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

The opposite of -\sqrt{3} is \sqrt{3}.

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

The opposite of -2 is 2.

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

Combine \sqrt{3} and \sqrt{3} to get 2\sqrt{3}.

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

Subtract 2 from 2 to get 0.

$$2\sqrt{3}$$