Like radicals are radical expressions that share the same index (the type of root) and the same radicand (the number or expression under the root symbol). When both of these parts match, you can add or subtract the radicals by combining their coefficients, just like combining like terms in algebra.
The Two Parts That Must Match
Every radical expression has two defining components. The index tells you what kind of root you’re taking: a square root has an index of 2, a cube root has an index of 3, and so on. The radicand is whatever sits under the root symbol. In √5, the index is 2 and the radicand is 5. In ∛7, the index is 3 and the radicand is 7.
For two radicals to be “like,” both pieces must be identical. So 3√6 and 5√6 are like radicals because they’re both square roots of 6. But √6 and ∛6 are not like radicals, even though the radicand is the same, because one is a square root and the other is a cube root. Similarly, √5 and √7 are not like radicals because their radicands differ.
Why Like Radicals Matter
The concept exists for one practical reason: you can only add or subtract radicals when they’re like. This works exactly the way combining like terms does in basic algebra. If you can add 4x + 9x to get 13x, you can add 4√x + 9√x to get 13√x. The radical part stays the same, and you only combine the coefficients (the numbers in front).
A few examples make this concrete:
- Addition: 2√6 + 5√6 = 7√6. You add the coefficients (2 + 5) and keep √6 unchanged.
- Subtraction: 4√10 − 5√10 = −√10. The coefficients give you 4 − 5 = −1, so the result is −1√10, or simply −√10.
- Unlike radicals: 3√2 + 4√3 cannot be simplified further. The radicands are different, so these terms stay separate.
Simplifying to Reveal Like Radicals
Sometimes two radicals look unlike at first glance but turn out to be like radicals once you simplify them. This is one of the most useful skills in working with radical expressions. The key is to break the radicand into factors, pulling out any perfect squares (or perfect cubes, for cube roots).
For example, suppose you need to add √20 + √5. These look unlike because 20 and 5 are different radicands. But √20 can be simplified: factor 20 into 4 × 5, then separate the root into √4 · √5, which equals 2√5. Now the problem becomes 2√5 + √5 = 3√5. The radicals were like all along; you just had to simplify first to see it.
This step is easy to skip, so it’s worth building the habit of simplifying every radical before deciding whether terms can be combined.
A Common Mistake to Avoid
The most frequent error with like radicals is adding the radicands instead of the coefficients. If you see 2√6 + 5√6 and write √13 or 7√12, that’s incorrect. The radical part never changes during addition or subtraction. Only the coefficients combine. The correct answer is 7√6.
Think of it this way: the radical symbol acts like a variable. You wouldn’t add 2x + 5x and get 7x². The x stays the same, and the numbers in front get added. Radicals follow the same logic.
Expressions With Multiple Terms
When you’re working with longer expressions that mix different radicals, group the like radicals together before combining. Take this example:
(5√x − 4√y) − (4√x − 7√y)
First, distribute the negative sign across the second group: 5√x − 4√y − 4√x + 7√y. Then rearrange by like radicals: the √x terms give you 5√x − 4√x = √x, and the √y terms give you −4√y + 7√y = 3√y. The final answer is √x + 3√y.
How Multiplication and Division Differ
Unlike addition and subtraction, multiplication and division do not require like radicals. You can multiply two radicals as long as they share the same index, even if their radicands are completely different. The product rule lets you combine them under a single root: √A · √B = √(A · B). The same idea applies to division: √A ÷ √B = √(A/B).
So while √3 + √5 can’t be simplified, √3 · √5 can be written as √15. This distinction trips up a lot of students. The “like” requirement applies only when you’re adding or subtracting.