# Denominator

## Definition

The denominator is the bottom number in a fraction. We write the denominator just below the fraction bar. For example, in the fraction ${\dfrac{4}{5}}$, 5 is the denominator.

The denominator acts as the divisor in a fraction. It divides the value above the fraction bar which is the numerator.

Besides numbers, denominators can also be variables such as y in ${\dfrac{x}{y}}$, and q in ${\dfrac{p}{q}}$

The denominator plays a significant role in adding, subtracting, or comparing two or more different fractions.

## Common Denominator

When two or more fractions have the same denominator, then the denominators are called common denominators.

For example, ${\dfrac{5}{9}}$ and ${\dfrac{2}{9}}$, are two fractions having the same denominator 9.

### Least Common Denominator

The least or lowest common denominator (LCD) means the lowest common multiple of the denominators that we get for two or more fractions with no common denominators.

If we have ${\dfrac{1}{2}}$, ${\dfrac{3}{4}}$, and ${\dfrac{7}{8}}$, then what will be the LCD?
The lowest common multiple of 2, 4, and 8 is 8. So the LCD for ${\dfrac{1}{2}}$, ${\dfrac{3}{4}}$, and ${\dfrac{7}{8}}$ is 8.

Find the least common denominator for ${\dfrac{2}{3}}$, ${\dfrac{4}{9}}$, and ${\dfrac{11}{12}}$.

Solution:

Here, the denominators are 3, 9, and 12.
∴ lowest common multiple of 3, 9, and 12 = 36
Thus, the least common denominator of the fractions ${\dfrac{1}{2}}$, ${\dfrac{3}{4}}$, and ${\dfrac{7}{8}}$ is 36.

## How to Rationalize the Denominator

We rationalize the denominator by moving the root from the denominator to the numerator. To simplify, we follow this process to make the denominator rational.

Suppose we have a fraction:

${\dfrac{2}{\sqrt{5}}}$

We will multiply the term with root both with the numerator and denominator

${=\dfrac{2}{\sqrt{5}}\times \dfrac{\sqrt{5}}{\sqrt{5}}}$

${=\dfrac{2\sqrt{5}}{\left( \sqrt{5}\right) ^{2}}}$

${=\dfrac{2\sqrt{5}}{5}}$

Rationalize ${\dfrac{3}{2\sqrt{3}}}$

Solution:

As we know, we have to move the root from the denominator to the numerator,
∴ Multiplying the root with both numerator and denominator,
${=\dfrac{3}{2\sqrt{3}}\times \dfrac{\sqrt{3}}{\sqrt{3}}}$
${=\dfrac{3\sqrt{3}}{2\times 3}}$
${=\dfrac{\sqrt{3}}{2}}$
∴ The rational form of the fraction ${\dfrac{3}{2\sqrt{3}}}$ is ${=\dfrac{\sqrt{3}}{2}}$.