Simplest Form — Definition, Formula & Examples

Simplest form is when a fraction or expression has been reduced as far as possible, meaning no common factor (other than 1) is shared by the numerator and denominator.

A fraction $\frac{a}{b}$ is in simplest form (also called lowest terms) when $\gcd(a, b) = 1$ , where $\gcd$ denotes the greatest common divisor. More broadly, an algebraic expression is in simplest form when no like terms can be combined and no further factoring or cancellation can be performed.

How It Works

To put a fraction in simplest form, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by that number. For algebraic expressions, combine like terms first, then factor and cancel any common factors. If the GCD is 1, the fraction is already in simplest form.

Worked Example

Problem: Write the fraction 18/24 in simplest form.

Step 1: Find the GCD of 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The greatest common factor is 6.

\gcd(18, 24) = 6

Step 2: Divide both the numerator and the denominator by 6.

\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Answer: The simplest form of 18/24 is

\frac{3}{4}

Why It Matters

Simplifying fractions and expressions is a routine step in nearly every algebra and pre-algebra problem. Answers left unsimplified are typically marked incomplete on tests. In later courses like calculus, working with simplest form prevents errors when solving equations or evaluating limits.

Common Mistakes

Mistake: Dividing numerator and denominator by a common factor that is not the greatest common factor, then stopping too early.

Correction: After dividing, check whether the new numerator and denominator still share a common factor greater than 1. Repeat until the GCD is 1, or find the GCD at the start to do it in one step.