And Subtracting Square Roots Worksheet

Working With Radical Expressions That Involve Addition and Subtraction

Most students get tripped up on this because they think you can combine any two square roots. You can't. The problem gets worse when worksheets dump twelve problems on a page and students panic and start just adding the numbers under the radicals anyway. I've seen it constantly.

The And Subtracting Square Roots Worksheet Problem

A typical worksheet will give you something like 50 + 18 - 8 and expect you to simplify it to a single number. The trick is that none of those individual radicals are in their simplest form yet. You have to break each one down first by factoring out perfect squares. So 50 becomes 52, 18 becomes 32, and 8 becomes 22. Once they all share that 2 term, you just combine the coefficients: 5 + 3 - 2 = 6, so the answer is 62. The moment the radicands don't match after simplification, the expression stops. You can't merge 3 and 5 any more than you can merge apples and oranges. That's usually where people lose points. They forget to simplify first and just stare at the original form. I ran into this with a student last year who was working through a practice sheet. They had a problem like 75 - 27 + 48 and they tried to just subtract 75 minus 27 and then add 48 under one radical. They ended up with 96 and called it done. We sat with it for about ten minutes and I just pointed at the three separate terms and asked them what each one simplified to. Once they saw they were all multiples of 3, it clicked. That's the whole thing really. Factor out the perfect squares until every term reveals what it's actually made of.

Step-by-step breakdown

Step one: Identify every radical in the expression. Write them all out separately so you can see what you're working with. Step two: Factor each radicand into a perfect square times whatever is left. For example, 72 breaks into 36 × 2. You already know that 36 is a perfect square, so 72 becomes 62. Do this for every single term in the expression. Step three: Rewrite the full expression using the simplified forms. This is where most students skip ahead and make mistakes, so slow down here.

Step four: Group terms that have identical radicands. Only those terms can be combined through addition or subtraction. Terms with different radicands stay separate. Step five: Combine the coefficients and attach the common radical. If a term has no visible coefficient, it's 1. So 2 is actually 12. Forgetting the implicit 1 is a surprisingly common error. Step six: Check your final answer. Make sure nothing inside a remaining radical can be simplified further. If you find a perfect square factor hiding in there, go back and reduce it.

Common pitfalls to watch for

The biggest issue is not simplifying before combining. You'll see students take 20 + 5 and just write 25 = 5, which is wrong. 20 simplifies to 25, so the correct answer is 35. The numbers inside the radicals don't add together directly. Another frequent mistake involves subtraction and sign errors. When you're combining coefficients, keep track of whether you're adding or subtracting each one. A negative coefficient in front of a term means you're subtracting from the total. Mixing this up flips your answer entirely. Some worksheets include terms that look similar but aren't. 12 and 18 both simplify to multiples of 3, but 12 = 23 and 18 = 33, so they do combine. However, 12 and 15 don't combine because 15 doesn't contain a perfect square factor beyond 1. Students sometimes assume anything close in value combines easily.

When this approach breaks down

If you end up with a mix of radicals that don't share any common factors after simplification, the expression is already in its simplest form. You can't force a combination. This happens more often than students expect, especially on extended practice sheets that include terms like 7, 11, or 13 alongside simplified terms. Another limitation shows up when the problem involves nested radicals or rational denominators. Standard addition and subtraction worksheets usually avoid these, but if you encounter them, you'll need additional techniques like rationalizing the denominator or denesting the radical first. For practice material, search for "And Subtracting Square Roots Worksheet" on math education sites. Many offer printable PDFs with answer keys. I usually recommend the ones with about ten to fifteen problems that gradually increase in difficulty. The ones with thirty problems in a block tend to burn students out without adding much value.

Quick reference for perfect squares

Knowing your perfect squares up to 144 speeds this process significantly. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. When you see a radicand, scan for the largest perfect square factor first. Factor that out, take the square root of the perfect square, and move the result outside the radical. Repeat until the remaining radicand has no perfect square factors other than 1.