The actual process
You can only add radicals together when their radicands and indices match after simplification. That means 35 + 25 = 55, and nothing more complicated than that without some intermediate steps. Most of the mistakes I see come from people trying to combine terms that look similar but aren't actually like terms. They'll write 12 + 3 = 15, which is wrong because they skipped the simplification step. Here's the method. First, simplify every radical individually. Break down the radicand into factors where one is a perfect square (for square roots), a perfect cube (for cube roots), or whatever power matches your index. Then check if any simplified terms share the same radicand. If they do, add the coefficients and keep the radical part unchanged. If they don't, the expression is already in its simplest form.
How To Add Radicals
Let me walk through something that trips people up constantly. Say you need to simplify 218 + 58. You can't just add them directly because the radicands are different. You simplify each one separately. 18 breaks into 9 × 2, which gives you 32. 8 breaks into 4 × 2, which gives you 22. Now both terms have 2, so you add the coefficients: 2(32) + 5(22) = 62 + 102 = 162. That's it. The radical part never changes during addition. I once had a student who was working with cube roots and got completely stuck on ³16 + ³54. They wanted to just add the radicands and get ³70. I told them to factor out the perfect cubes first. ³16 = ³8 × ³2 = 2³2. ³54 = ³27 × ³2 = 3³2. So the sum is 5³2. The same principle applies regardless of the index. You're always looking for perfect nth powers inside the radicand. There's a particular edge case that always catches people off guard. You'll see something like (x²) + x and think you can factor out x to get (x + 1)x. That's wrong. (x²) equals |x|, not x, and unless you know x is non-negative, you can't just drop the absolute value. I learned this the hard way when I was tutoring someone who lost points on a test for exactly this mistake. They wrote (x²) + x = xx + x and then combined it into (x+1)x without considering that x could be negative. The correct answer depends on the domain you're working in.
Another thing worth noting: radicals with different indices can't be added directly either. ²3 + ³3 isn't simplifiable in any clean way. You could convert everything to rational exponents and get 3^(1/2) + 3^(1/3), but that's not really an improvement. In practice, you just leave it as is. Some textbooks will ask you to find a common index using the least common multiple of the indices, but that rarely produces a cleaner result. For ²3 + ³3, the LCM of 2 and 3 is 6, so you'd rewrite it as 27 + 9. That's technically a single expression but it's not simpler in any meaningful sense. One counter-intuitive point that beginners miss: sometimes the radical part disappears entirely during addition. Take 12 - 3. You simplify 12 to 23, then 23 - 3 = 3. The coefficients subtract to give you a smaller number. Or consider (5 + 3)(5 - 3), which equals 5 - 3 = 2. The radicals cancel out completely. This happens often enough in textbook problems that students sometimes assume every addition problem will leave a radical behind, which it won't. The real limitation of this whole approach is that it only works when radicals can be simplified to like terms. If you have something like 2 + 3 + 5, there's no simplification possible. These are as far as you're going to get. Don't waste time looking for a pattern that isn't there. The same goes for nested radicals like (2 + 3), which generally require a completely different technique to denest, and that's not related to addition at all.
Get the Full Details
One more practical thing: when you're working with variables under the radical, always check whether the problem specifies that variables represent positive real numbers. If it doesn't, you need to be careful about assumptions. (x) = x², which is fine, but (x²) = |x|, and that absolute value matters if x could be negative. I've seen professional engineers make this assumption in applied work and get answers that are off by a sign. It's a small thing but it compounds. If you're working through problems and getting stuck, the first thing to check is whether each radical is fully simplified. Eight out of ten errors come from skipping that step. Factor the radicand completely, pull out every perfect power you can, and then compare the remaining radical parts. If they're identical, combine the coefficients. If they're not, you're done.
