Adding and Subtracting Radicals
Simple Explanation
Radicals can be added or subtracted directly only when they are "like radicals" — the same index and the same radicand. Combine them the way you combine like terms, by adding or subtracting their coefficients and keeping the radical part unchanged.
Why Do We Need It?
Recognizing like radicals (sometimes after simplifying first) is what makes it possible to combine terms in longer algebraic expressions involving roots.
Worked Example
Add radicals after simplifying
Simplify √8 + √18.
Why Does This Work?
Like radicals represent the same "unit" (the same irrational number), so combining their coefficients is exactly the same distributive-property idea as combining like terms: 2√2 + 3√2 = (2+3)√2.
Real-Life Example
Combining diagonal distances
A delivery route covers two diagonal shortcuts, one of length √8 km and another of length √18 km.
Simplifying and combining gives a total of 5√2 km ≈ 7.07 km, a single clean number for route planning instead of two separate unsimplified radicals.
Practice
Simplify 5√3 − 2√3.
EasyCommon mistake
Trying to combine √a + √b into √(a+b) — radicals do NOT add this way (e.g. √4 + √9 = 2+3 = 5, but √13 ≈ 3.6). Only like radicals combine, and only by adding their coefficients.
Quick Review
- Only "like radicals" — same index, same radicand — can be added or subtracted directly.
- Combine like radicals by adding/subtracting their coefficients: a·ⁿ√x ± b·ⁿ√x = (a±b)·ⁿ√x.
- Simplify each radical first — unlike-looking radicals may become like radicals after simplifying.