Simplifying Radicals
Simple Explanation
A radical is fully simplified when the radicand has no more perfect-nth-power factors left inside it. Simplify by splitting the radicand into a perfect-power factor and a leftover factor, then pulling the perfect power out using the product rule.
Why Do We Need It?
A simplified radical is easier to compare, add, and work with — √50 and √2 look unrelated until √50 is simplified to 5√2, revealing they share the same radical part.
See It
A number line marking the approximate positions of √2, √50, and 5√2 to show they represent related values
Formula
Product Rule for Radicals
ⁿ√a · ⁿ√b = ⁿ√(ab)
The product of two radicals with the same index equals the radical of the product of their radicands.
- a, b
- — the radicands (a, b ≥ 0 when n is even)
- n
- — the common index (root) of both radicals
When to use it: Whenever you simplify a radical by factoring out a perfect nth-power factor, or multiply two radicals with the same index.
Worked Example
Simplify a square root
Simplify √50.
Why Does This Work?
The product rule for radicals says √(ab) = √a · √b, so factoring out a perfect square and applying the rule "extracts" its exact square root, leaving only the non-perfect-square part under the radical.
Real-Life Example
Simplifying diagonal-length measurements
A carpenter calculates the diagonal of an 8-by-2 rectangular board using the Pythagorean theorem, getting √68.
Simplifying √68 = 2√17 gives a shorter, more useful form for further calculation or for reading off a tape measure alongside a decimal approximation.
Practice
Simplify √18.
EasySimplify √75.
MediumCommon mistake
Stopping at a factor that is not actually the largest perfect square (e.g. writing √50 = √2 · √25 correctly evaluated, but starting from a smaller perfect square like 50=... and leaving an unsimplified radical) — always check no further perfect-square factor remains.
Quick Review
- Simplify by factoring out the largest perfect-nth-power factor of the radicand.
- ⁿ√(ab) = ⁿ√a · ⁿ√b lets you pull that perfect power out of the radical.
- A radical is fully simplified only when no perfect-power factor remains inside it.