Math

Add & Subtract Radical Expressions Calculator

Simplify and combine radical expressions step-by-step. Add and subtract square roots, cube roots, and higher-order radicals with detailed solutions.

This calculator simplifies each radical, identifies like radicals, and combines them. Enter expressions in the form: coefficient√radicand (e.g., 3√8) or ∛radicand for cube roots. Use a plus or minus sign to indicate the operation between terms. The calculator shows all simplification steps.
radical-expressions-calc
Enter square root expressions
Format: coefficient√radicand, separated by + or −. Example: 3√8 + 2√18 − √2
Simplified result

Simplification examples

Below are step-by-step examples of how the calculator simplifies and combines radicals:

Example 1: √18 + √8

Step 1: Simplify √18 18 = 9 × 2 = 3² × 2 √18 = √(3² × 2) = 3√2 Step 2: Simplify √8 8 = 4 × 2 = 2² × 2 √8 = √(2² × 2) = 2√2 Step 3: Identify like radicals 3√2 and 2√2 both have √2, so they are like radicals Step 4: Combine 3√2 + 2√2 = (3 + 2)√2 = 5√2 Answer: 5√2

Example 2: 2∛16 + ∛54

Step 1: Simplify 2∛16 16 = 8 × 2 = 2³ × 2 ∛16 = ∛(2³ × 2) = 2∛2 So 2∛16 = 2 × 2∛2 = 4∛2 Step 2: Simplify ∛54 54 = 27 × 2 = 3³ × 2 ∛54 = ∛(3³ × 2) = 3∛2 Step 3: Identify like radicals 4∛2 and 3∛2 both have ∛2, so they are like radicals Step 4: Combine 4∛2 + 3∛2 = (4 + 3)∛2 = 7∛2 Answer: 7∛2

Perfect powers reference

Use this table to identify perfect powers when simplifying radicals:

Perfect squares Perfect cubes Perfect 4th powers
1, 4, 9, 16, 25 1, 8, 27, 64, 125 1, 16, 81, 256
36, 49, 64, 81, 100 216, 343, 512, 729 625, 1296, 2401

How the calculator works

The calculator follows these steps to simplify and combine radical expressions:

Step 1: Parse input Extract each term (coefficient, index, radicand) Step 2: Simplify each radical Factor the radicand into perfect powers and remaining factors Extract the perfect power outside the radical Step 3: Identify like radicals Group terms with the same index and radicand Step 4: Combine like radicals Add/subtract coefficients of like radicals Keep the radical part unchanged Step 5: Display result Show the simplified expression and step-by-step work Formula for simplifying: If radicand = (perfect power) × (remaining) Then index√radicand = (index√(perfect power)) × (index√(remaining)) And index√(perfect power) simplifies to a whole number

Common questions

  • A radical expression contains a root symbol (√, ∛, ⁴√, etc.). The number inside the root is the radicand; the number outside the root symbol is the coefficient. For example, in 3√8, the coefficient is 3, the index is 2 (square root), and the radicand is 8. Radical expressions follow the same addition and subtraction rules as variables: you can only combine like radicals (same index and radicand).
  • Like radicals have the same index (root type) and the same radicand (number under the root). For example, 2√5 and 7√5 are like radicals because both have index 2 and radicand 5. You can add or subtract them: 2√5 + 7√5 = 9√5. But 2√5 and 2√3 are unlike radicals (different radicands), so you cannot combine them—the result stays as 2√5 + 2√3.
  • Simplify by factoring out perfect powers. For √8: factor 8 = 4 × 2 = 2² × 2, so √8 = √(2² × 2) = 2√2. For ∛54: factor 54 = 27 × 2 = 3³ × 2, so ∛54 = 3∛2. The key is finding perfect square (or cube, etc.) factors. Once you extract the perfect power from under the radical, you can combine like radicals.
  • √2 and √3 are unlike radicals—they have different radicands. Just as you cannot add 2x + 3y directly (they're different variables), you cannot combine √2 + √3. They must stay separate in your final answer. However, if you simplify radicals first, sometimes they become like radicals. For example, √8 + √2 simplifies to 2√2 + √2 = 3√2 because √8 simplifies to 2√2.
  • The calculator finds the prime factorization of the radicand, then extracts perfect powers matching the index. For a square root (index 2), it looks for pairs of factors; for a cube root (index 3), it looks for triples. For example, to simplify ⁴√80: factor 80 = 16 × 5 = 2⁴ × 5, so ⁴√80 = 2 × ⁴√5 = 2⁴√5. The calculator shows each step.
  • That's common and the whole point of simplifying first. For example, √18 + √8. Simplify: √18 = 3√2 and √8 = 2√2. Now they're like radicals! Combine: 3√2 + 2√2 = 5√2. The calculator automatically simplifies each radical, identifies like terms, and combines them.
  • No. Even though the radicand is the same (5), different indices (2 vs 3) mean they're unlike radicals. √5 (square root) and ∛5 (cube root) cannot be combined. They must stay separate: √5 + ∛5. Sometimes higher-index roots can be rewritten using fractional exponents (√5 = 5^(1/2), ∛5 = 5^(1/3)), but they still cannot be added directly.
  • The coefficient is the number multiplied in front of the radical. In 5√7, the coefficient is 5. When you combine like radicals, you add/subtract the coefficients. For example, 3√2 + 5√2 = (3+5)√2 = 8√2. The radical part (√2) stays the same; only the coefficient changes. If there's no visible coefficient, it's 1 (e.g., √3 is the same as 1√3).
  • Subtraction works the same as addition—you can only subtract like radicals. 7√5 - 3√5 = 4√5. The radicands must match. If you have unlike radicals, simplify first to see if they become like. For example, √20 - √5: simplify √20 = 2√5, so 2√5 - √5 = √5. Subtraction of negative coefficients works too: if the result is negative, write it normally, e.g., 3√2 - 7√2 = -4√2.
  • Rationalizing removes radicals from the denominator of a fraction. For example, 1/√2 becomes √2/2 by multiplying top and bottom by √2. This calculator focuses on adding/subtracting radicals in their simplified form; if your final answer has a denominator with a radical, multiply by the conjugate or simplifying radical to clear it. This is often a follow-up step.
  • Yes. You can enter negative coefficients (e.g., -3√5) directly. The calculator tracks signs correctly. For example, -2√3 + 5√3 = 3√3. If the result is negative (e.g., 3√2 - 7√2), it displays as -4√2. Subtraction also works: 5√2 - 3√2 = 2√2, or 2√8 - √18 simplifies to 4√2 - 3√2 = √2.
  • If the coefficient is 1, it's typically written as just the radical. For example, if you calculate 5√2 - 4√2, the result is 1√2, which is written as √2. The calculator shows both the numerical coefficient (1) and the simplified form (√2). A zero result (e.g., 3√5 - 3√5 = 0) is written as 0.