Radical Simplifier Calculator
Enter a whole number (positive integer). This tool simplifies square root expressions only -- for general roots use the Radical Calculator.
How to use this radical simplifier
- Enter a positive whole number (integer) in the input field above.
- The simplified radical form appears instantly -- for example, sqrt(72) becomes 6sqrt(2).
- The step-by-step section shows which perfect square factor was used and how the product rule was applied.
- Check whether the number is already in simplest form or is a perfect square (exact integer).
- Click Copy to grab the simplified form for use in your homework, equation, or spreadsheet.
How to simplify radical expressions -- the complete method
Step 1: find the largest perfect square factor
The first and most important step is factoring the radicand into a perfect square and a remainder. For sqrt(180): check which perfect squares (4, 9, 16, 25, 36...) divide 180 evenly. Both 4 and 36 divide 180 evenly (180/4 = 45, 180/36 = 5). Always use the largest perfect square factor -- 36 in this case -- to simplify in one step rather than multiple passes.
Step 2: apply the product rule
The product rule states: sqrt(a x b) = sqrt(a) x sqrt(b). Once you have identified the largest perfect square factor, apply this rule. For sqrt(180): sqrt(180) = sqrt(36 x 5) = sqrt(36) x sqrt(5) = 6 x sqrt(5). The result is written as the coefficient (6) multiplied by the remaining radical (sqrt(5)).
Step 3: verify the result
Always verify by squaring the simplified form and checking it equals the original radicand. For 6sqrt(5): (6)^2 x 5 = 36 x 5 = 180. Confirmed. The coefficient squared times the remaining radicand must equal the original radicand every time. This check catches errors in factoring and ensures simplest form.
Perfect square reference table for simplification
| Radicand | Largest PSF | Factor form | Simplified |
|---|---|---|---|
| sqrt(12) | 4 | sqrt(4 x 3) | 2sqrt(3) |
| sqrt(18) | 9 | sqrt(9 x 2) | 3sqrt(2) |
| sqrt(20) | 4 | sqrt(4 x 5) | 2sqrt(5) |
| sqrt(27) | 9 | sqrt(9 x 3) | 3sqrt(3) |
| sqrt(32) | 16 | sqrt(16 x 2) | 4sqrt(2) |
| sqrt(45) | 9 | sqrt(9 x 5) | 3sqrt(5) |
| sqrt(50) | 25 | sqrt(25 x 2) | 5sqrt(2) |
| sqrt(72) | 36 | sqrt(36 x 2) | 6sqrt(2) |
| sqrt(75) | 25 | sqrt(25 x 3) | 5sqrt(3) |
| sqrt(98) | 49 | sqrt(49 x 2) | 7sqrt(2) |
| sqrt(108) | 36 | sqrt(36 x 3) | 6sqrt(3) |
| sqrt(180) | 36 | sqrt(36 x 5) | 6sqrt(5) |
What is simplest radical form?
A square root expression is in simplest radical form when three conditions are all met. First, the radicand (number under the root symbol) has no perfect square factors other than 1. Second, the radicand is a whole number -- no fractions under the radical. Third, no radical expressions appear in the denominator of a fraction (rationalized form). This calculator handles the first condition, which is the most common simplification task in algebra.
Examples of expressions in simplest radical form: sqrt(2), sqrt(3), sqrt(5), sqrt(6), sqrt(7), 2sqrt(3), 5sqrt(2), 4sqrt(7). Examples NOT in simplest form: sqrt(8) (contains 4 = 2^2), sqrt(12) (contains 4), sqrt(18) (contains 9), sqrt(50) (contains 25). For sqrt(50): 50 = 25 x 2, so sqrt(50) = 5sqrt(2).
Simplest radical form is required in algebra, trigonometry, precalculus, and calculus courses because it makes expressions easier to compare, add, subtract, and use in proofs. Two radical expressions can only be added directly if they have the same radicand: 3sqrt(2) + 5sqrt(2) = 8sqrt(2). If radicands differ, you cannot combine them until both are in simplest form -- then you can check whether they match.
- sqrt(8) = 2sqrt(2) (8 = 4 x 2, largest PSF = 4)
- sqrt(12) = 2sqrt(3) (12 = 4 x 3, largest PSF = 4)
- sqrt(50) = 5sqrt(2) (50 = 25 x 2, largest PSF = 25)
- sqrt(7) = sqrt(7) (already simplest, no perfect square factor)
Interpreting your simplification result
Perfect square -- exact integer result
When the radicand is a perfect square (1, 4, 9, 16, 25...), the result is an exact integer with no radical remaining. sqrt(144) = 12 exactly. The coefficient becomes the result and the remaining radicand becomes 1, meaning there is no radical symbol left. The decimal equals the coefficient exactly.
Already in simplest form
When the radicand has no perfect square factors, it is already in simplest form -- the coefficient is 1 and the radicand is unchanged. sqrt(7), sqrt(11), sqrt(13), sqrt(14) are already simplified. The decimal result is still irrational (non-terminating), but no further symbolic simplification is possible.
Simplified to coefficient x sqrt(remainder)
When a perfect square factor greater than 1 exists, the result is a coefficient times a remaining radical. sqrt(72) = 6sqrt(2) -- coefficient 6, remaining radicand 2. The remaining radicand is always in simplest form (no further perfect square factors). Verify: 6^2 x 2 = 72.
Quick radical simplification tips
- Always use the LARGEST perfect square factor first: pulls out the maximum coefficient in one step. Using 4 instead of 36 for sqrt(180) requires a second pass.
- Even numbers to check first: 4 divides all multiples of 4; 9 divides digit-sum multiples of 9; 25 divides numbers ending in 0 or 5 that are divisible by 25.
- Perfect squares to memorise: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Recognising these instantly speeds up every simplification.
- Verify always: square the coefficient, multiply by remaining radicand, confirm it equals the original. Catches all factoring mistakes.
- Cube root simplification: same method but look for perfect cube factors (8, 27, 64, 125...) instead of perfect square factors.
Common radical simplification mistakes
Not using the largest perfect square factor
Using a smaller perfect square factor produces a partially simplified result that still needs further work. For sqrt(72): if you factor out 4 instead of 36, you get 2sqrt(18), but sqrt(18) = 3sqrt(2), so the final answer should be 6sqrt(2). You get there eventually, but using the largest factor (36) gives 6sqrt(2) directly. Always check for larger factors before committing to a partial simplification.
Misapplying the product rule to addition
The product rule works for multiplication only: sqrt(a x b) = sqrt(a) x sqrt(b). It does NOT apply to addition: sqrt(a + b) is NOT equal to sqrt(a) + sqrt(b). This is an extremely common error. sqrt(9 + 16) = sqrt(25) = 5, NOT sqrt(9) + sqrt(16) = 3 + 4 = 7. Always simplify what is INSIDE the radical before trying to split it.
Leaving a non-simplified radical in an answer
When an exam or textbook asks for "simplified form," leaving sqrt(50) as sqrt(50) is incorrect even though it is technically valid notation. The expected answer is 5sqrt(2). Teachers and graders look for simplest radical form. This calculator flags whether simplification is possible and shows the complete simplified result.
Forgetting to verify
After simplifying, always verify: coefficient^2 x remaining radicand = original radicand. For 6sqrt(2): 6^2 x 2 = 36 x 2 = 72. If the check fails, you made a factoring error. Verification takes five seconds and catches every mistake. This calculator shows the verification in the step-by-step section automatically.