# How do you rationalize the denominator and simplify 3/(2sqrt3)?

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

#### Explanation:

We have

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

Right now, the denominator is irrational - the $\sqrt{3}$ is an irrational number and so we'd like that out of there (when we think of fractions in terms of pizza, we can think of the numerator as the number of slices and the denominator as determining the size of the slices. When dealing with a fraction that has an irrational number in it, we prefer to have it in the numerator so that while the number of slices is irrational, the size of the slices is not).

We can use the rule $\sqrt{x} \sqrt{x} = x$:

$\frac{3}{2 \sqrt{3}} \left(1\right) = \frac{3}{2 \sqrt{3}} \left(\frac{\sqrt{3}}{\sqrt{3}}\right) = \frac{3 \sqrt{3}}{2 \sqrt{3} \sqrt{3}} = \frac{3 \sqrt{3}}{2 \times 3} = \frac{3 \sqrt{3}}{6}$

We can then simplify:

$\frac{3 \sqrt{3}}{6} = \frac{3 \sqrt{3}}{3 \times 2} = \left(\frac{3}{3}\right) \left(\frac{\sqrt{3}}{2}\right) = \left(1\right) \left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}$
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