How To Add And Subtract Radicals | Clean Steps, No Confusion

Radicals can feel slippery at first because the math looks “unfinished.” A square root like √8 is not a tidy whole number, yet it still behaves like a term you can simplify, group, and combine. Once you know what counts as a “match,” adding and subtracting radicals becomes as routine as combining like terms in algebra.

This article shows a repeatable way to do it. You’ll learn how to spot like radicals, how to simplify first so matches reveal themselves, and how to avoid the classic traps that cost points on tests.

What Radicals Mean In Algebra

A radical sign (√ ) tells you to take a root. The most common is the square root, written √a, which means “the number that squares to a,” using the principal (non-negative) root for real numbers. You’ll see cube roots (∛a) and higher roots too. The small number on the radical, called the index, is 2 for square roots (often not written), 3 for cube roots, and so on.

In expressions, a radical can act like a term. Think of 5√2 as “5 times √2.” That viewpoint makes the add/subtract rule feel natural: you can combine only when the radical part matches exactly, the same way you can combine 5x and 3x but not 5x and 3y.

When Two Radicals Are “Like”

Two radicals are like radicals when they share the same index and the same radicand (the value under the radical). So √2 and 7√2 are like. √2 and √8 are not like at the start, yet √8 can simplify to 2√2, which turns them into a match.

This single idea drives nearly every problem: simplify each radical as far as it will go, rewrite the expression, then combine only the truly like terms.

How To Add And Subtract Radicals Step By Step

Use this routine every time. It stays steady whether you’re working with numbers, variables, or higher roots.

Step 1: Simplify Each Radical Term

Simplifying means pulling perfect powers out from under the radical. With square roots, you pull out perfect squares. With cube roots, you pull out perfect cubes.

• Square roots: √(ab) = √a · √b, so if a is a perfect square, it can move outside as a whole number.
• Cube roots: ∛(ab) = ∛a · ∛b, so if a is a perfect cube, it can move outside as a whole number.

Example: √72 = √(36·2) = 6√2. The 36 leaves the radical because √36 = 6.

Step 2: Rewrite Using Standard Form

Write each term as (coefficient)·(simplified radical). For square roots, standard form often means no perfect square factor remains inside the radical.

Example: 3√8 becomes 3·(2√2) = 6√2.

Step 3: Group Like Radicals

After simplification, circle or underline terms with identical radical parts.

Example: 6√2 − √18 + 5√2. Since √18 = 3√2, the expression becomes 6√2 − 3√2 + 5√2, and all three are like.

Step 4: Add Or Subtract The Coefficients

Keep the radical part unchanged. Combine only the numbers in front.

Example: (6 − 3 + 5)√2 = 8√2.

Step 5: Final Check

Scan the result. If any radical still hides a perfect power factor, simplify again. If you expanded parentheses early, verify signs stayed correct.

Adding And Subtracting Radicals With Like Terms In One Line

Once terms are simplified, adding or subtracting is a single move: (a√m) ± (b√m) = (a ± b)√m. OpenStax states the same rule using “like radicals,” linking it directly to combining like terms. OpenStax section on adding and subtracting radical expressions shows the pattern with worked examples.

That rule works for square roots, cube roots, and higher roots, as long as the index and radicand match after simplification.

What To Do When Radicals Don’t Match Yet

Many exercises are designed so “unlike” radicals turn into like radicals after simplification. Your job is to rewrite before you combine.

Simplify Before You Decide

Take √50 + √8. At first glance, the radicands differ. Simplify each term: √50 = √(25·2) = 5√2, and √8 = √(4·2) = 2√2. Now they match, so the sum is 7√2.

Stop When Simplification Is Done

Try √12 + √3. √12 becomes 2√3. Now the expression is 2√3 + √3 = 3√3. No further change is possible because 3 has no perfect square factor beyond 1.

Don’t Combine Truly Unlike Radicals

√2 + √3 cannot become √5. Roots do not add that way. The clean result is √2 + √3, and that’s a valid final answer in simplified form.

Radical Simplification Patterns Worth Memorizing

Speed comes from pattern recognition. You don’t need a long list, just a feel for perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) and perfect cubes (1, 8, 27, 64, 125).

The table below gives common simplifications that show up in homework, quizzes, and entrance exams.

Original Radical Term	Simplified Form	What Was Pulled Out
√8	2√2	4 = 2²
√18	3√2	9 = 3²
√20	2√5	4 = 2²
√45	3√5	9 = 3²
√72	6√2	36 = 6²
∛54	3∛2	27 = 3³
∛128	4∛2	64 = 4³
√(12x²)	2x√3	4x² = (2x)²

Adding And Subtracting Radicals With Variables

Variables add one extra rule: √(x²) becomes |x| in full generality. In many algebra classes, problems assume x ≥ 0, so √(x²) is written as x. Your teacher’s expectations matter here, so follow the convention used in your course notes.

Match The Entire Radical Part

Take 2√(3x) + 7√(3x). The radical parts match already, so you combine coefficients: 9√(3x).

Now take √(12x) + √(27x). Simplify each term: √(12x) = √(4·3x) = 2√(3x). Also √(27x) = √(9·3x) = 3√(3x). Now the sum is 5√(3x).

Watch Powers Of Variables Inside The Radical

Try √(50x²) − 3√(8x²). Simplify: √(50x²) = √(25·2·x²) = 5x√2 (or 5|x|√2 under the absolute-value form). Next √(8x²) = √(4·2·x²) = 2x√2. Now subtract: 5x√2 − 3(2x√2) = 5x√2 − 6x√2 = −x√2.

Parentheses And Distribution With Radicals

Many expressions hide like radicals behind parentheses. Distribute first, then simplify, then combine.

Example: 2(√18 − √8) + (3√2 − √50)

• Distribute: 2√18 − 2√8 + 3√2 − √50
• Simplify: 2(3√2) − 2(2√2) + 3√2 − 5√2
• Combine: 6√2 − 4√2 + 3√2 − 5√2 = 0

That last line can feel surprising. It’s a good reminder that simplification can reveal cancellation that wasn’t visible at the start.

Higher Roots And Mixed Indices

Square roots and cube roots do not mix. √(2) and ∛(2) are different radical types, so they never combine, even when the radicand matches. The index is part of the “like radical” test.

Example: ∛(16) + ∛(54). Simplify first: ∛(16) = ∛(8·2) = 2∛2 and ∛(54) = ∛(27·2) = 3∛2. Now add: 5∛2.

Common Mistakes That Break Answers

Most wrong answers come from a small set of habits. Fix these and your accuracy jumps fast.

Adding Radicands Instead Of Coefficients

√a + √b ≠ √(a+b). Keep radicals separate unless they are like radicals after simplification.

Skipping Simplification

If you treat √8 and √2 as unlike and stop, you miss that √8 = 2√2. Always simplify first.

Losing A Negative Sign

When subtraction and parentheses mix, distribute the negative across every term inside.

Dropping Variable Rules

If your course uses √(x²)=|x|, don’t drop the absolute value. If your course assumes x ≥ 0, stay consistent with that assumption across the whole problem.

Decision Rules You Can Apply Mid-Problem

When you’re stuck, use the quick checks below. They work during homework and also under timed test pressure.

What You See	What To Do Next	Mini Result Pattern
Different radicands	Simplify each radical	√(ab) → √a·√b
Same radicand, same index	Combine coefficients	a√m ± b√m → (a±b)√m
Same radicand, different index	Keep terms separate	√m + ∛m stays as is
Variables squared inside	Pull out the variable factor	√(kx²) → x√k (or |x|√k)
Parentheses with radicals	Distribute, then simplify	c(√a ± √b) → c√a ± c√b
Negative outside parentheses	Flip each sign inside	-(p + q) → -p – q
Perfect power factor inside	Pull it out fully	√(36m) → 6√m

Practice Set With Answers

Try these without rushing. Write each intermediate step once, so you can spot where a slip happens.

Problems

1. √48 + 2√75
2. 5√12 − 3√27
3. 2√(18x) + √(8x)
4. 3(√50 − √8) + 4√2
5. ∛250 − 2∛54

Answers With Brief Work

1. √48 = 4√3 and √75 = 5√3, so 4√3 + 2(5√3) = 14√3.
2. √12 = 2√3 and √27 = 3√3, so 5(2√3) − 3(3√3) = 10√3 − 9√3 = √3.
3. √(18x) = 3√(2x) and √(8x) = 2√(2x), so 2(3√(2x)) + 2√(2x) = 8√(2x).
4. 3√50 = 3(5√2) = 15√2 and 3√8 = 3(2√2) = 6√2, so 15√2 − 6√2 + 4√2 = 13√2.
5. ∛250 = ∛(125·2) = 5∛2 and ∛54 = 3∛2, so 5∛2 − 2(3∛2) = −∛2.

Study Moves That Make Radicals Easier Next Time

If radicals still feel slow, train two skills: spotting perfect powers and rewriting fast. A few minutes of targeted practice beats re-reading notes for an hour.

• Write the first ten perfect squares and perfect cubes on a corner of your scratch paper before a quiz.
• Practice simplification only: take 15 numbers and simplify √n or ∛n without adding anything yet.
• Check one worked lesson that matches your current level. Khan Academy’s worked video on simplifying sums of radicals is a solid reference when you want to see clean steps without extra fluff. Khan Academy video on adding and simplifying radicals walks through a multi-term example.

Mini Checklist Before You Turn In An Answer

• Every radical term is simplified (no hidden perfect power factor remains).
• Only like radicals were combined (same index, same radicand).
• Signs stayed correct through distribution and subtraction.
• The final expression is in simplest radical form.