Adding and Subtracting Radical Expressions Calculator (Free Tool + Guide)

Adding and Subtracting Radical Expressions Calculator

Radicals only combine when they match. Miss that one rule, and an otherwise correct algebra problem falls apart. Our free adding and subtracting radical expressions calculator checks your work instantly and shows every step, so you can see exactly where a problem goes right or wrong.

How to Use the Calculator

The tool sits on the "Add & Subtract Radicals" tab, right next to the Radical Equation Solver, so you can switch between the two without leaving the page.

• Pick the right tab. Click Add & Subtract Radicals at the top if you're combining terms rather than solving for x.
• Type or tap in your expression. Use the input box directly, or tap the on-screen keys below it: √ for a square root, ∛ for a cube root, ⁴√ for a fourth root, or root if you need a custom index. The ^ key handles exponents, and ( ) group terms when a radicand has more than one number inside it.
• Build multi-term expressions with + and -. Since this calculator is built for combining radicals, chain terms together the same way you'd write them on paper, for example 5√12 - 2√3 + √27.
• Not sure what to enter? Load an example. The Load an example... dropdown in the top right fills the input box for you, and the 🎲 Generate example button below the keys pulls a fresh random problem if you just want to practice.
• Filter examples by difficulty. Under the Example Problems section, the Easy / Medium / Hard toggle changes the set of sample expressions shown, so you can start simple and work up to messier radicands.
• Hit Solve. The calculator simplifies every term, groups the like radicals, and returns the combined answer.
• Made a mistake typing? Use Back to delete the last character or Reset to clear the input box and start over. Pressing Enter on your keyboard solves the problem the same way clicking Solve does, and Esc resets it.

What Are Radical Expressions?

A radical expression is any expression that includes a root, most commonly a square root, written with the radical symbol √. Examples include √8, 3√5, and 2√12 + √3. The number under the radical is called the radicand, and any number sitting in front of it is the coefficient.

You'll run into radical expressions in algebra 1 and 2, geometry (especially with the Pythagorean theorem), and pretty much anywhere distance or area calculations produce a non-perfect square.

The Core Rule: Like Radicals Only

Here's the one rule that decides everything: you can only add or subtract radicals if they have the same index and the same radicand. That means √3 and √3 combine. √3 and √5 do not, no matter how much you'd like them to.

This works the same way combining like terms in algebra does. You wouldn't add 3x and 5y and call it 8xy, and you can't add 3√2 and 5√3 and call it 8√5 either. The radicands have to match exactly first.

Example of Like Radicals

3√7 + 2√7 = 5√7

Both terms share the radicand 7, so you add the coefficients (3 and 2) and keep the radical unchanged.

Example of Unlike Radicals

3√7 + 2√5 cannot be combined further. It stays as 3√7 + 2√5, because 7 and 5 are different radicands.

When Radicals Look Different But Aren't

This is where most mistakes happen. Two radicals can look completely different on the page and still be like radicals once simplified. Take √8 and √18. Neither matches at first glance, but simplify them:

• √8 = √(4 × 2) = 2√2
• √18 = √(9 × 2) = 3√2

Once simplified, both have a radicand of 2. Now they combine: 2√2 + 3√2 = 5√2.

This is exactly why our adding and subtracting radical expressions calculator simplifies every term before attempting to combine it. Skipping that step is the single most common reason students get a "cannot combine" answer wrong.

Step-by-Step: How to Add and Subtract Radical Expressions

1. Simplify each radical first. Break the radicand into a perfect square factor and a remaining factor.
2. Identify like radicals. Group terms that now share the same radicand and index.
3. Add or subtract the coefficients of the like radicals. Leave the radicand untouched.
4. Leave unlike radicals separate. Don't force a combination that isn't mathematically valid.
5. Write the final simplified expression, combined terms first, unlike terms listed after.

Worked Example

Simplify: 5√12 - 2√3 + √27

• 5√12 = 5(2√3) = 10√3
• √27 = √(9 × 3) = 3√3
• Now every term shares the radicand 3: 10√3 - 2√3 + 3√3
• Combine coefficients: 10 - 2 + 3 = 11
• Final answer: 11√3

Common Mistakes to Avoid

Radical Expressions vs. Radical Equations

A radical expression, like the ones on this page, is just a value you're simplifying, adding, or subtracting. A radical equation sets one radical expression equal to another and asks you to solve for a variable. If you're working with a full equation instead, our radical equation calculator walks through isolating the variable and checking for extraneous solutions step by step.

Why Use a Calculator Instead of Doing It by Hand?

Doing radical arithmetic by hand is a good way to learn the rule the first few times. After that, it's mostly a place where small arithmetic slips (a missed factor, a sign error) cost you the whole problem. A calculator that shows its steps gives you two things at once: a fast answer to check homework against, and a visible trail of the simplification so you can spot exactly where your own work diverged.

Frequently Asked Questions

Try It on Your Own Problem

Whether you're checking algebra homework, prepping for a geometry test, or just brushing up before a placement exam, the calculator above gives you a simplified answer and the full working in seconds. Enter your expression and see exactly how each radical breaks down.