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Simplify Square Root Calculator
Write any square root, cube root or nth root in simplest radical form a√b. The calculator finds the prime factorization, pulls the perfect powers out of the radical and shows each step, plus the decimal value.
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Leave as 1 for a plain root, e.g. 3 for 3√72
Enter your values to see the result.
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How to use the Simplify Square Root Calculator
- Enter the radicand, the whole number under the root sign.
- Add a coefficient if there is a number in front of the radical.
- Choose the index: 2 for a square root, 3 for a cube root and so on.
- Read the simplified form and follow the steps.
How to simplify a radical
- Write the radicand as a product of primes: 72 = 2³ × 3².
- For an index k, split each prime power into groups of k: 2³ = 2² × 2 and 3² = 3².
- Each complete group leaves the radical as one factor: 2 × 3 = 6 outside.
- Whatever does not form a full group stays inside: 2.
- So √72 = 6√2. Multiply by any coefficient already in front.
The rule behind it is ⁿ√(a × b) = ⁿ√a × ⁿ√b for non-negative a and b, together with ⁿ√(aⁿ) = a.
Simplified square roots from 1 to 100 (non-perfect squares)
| √8 = 2√2 | √12 = 2√3 | √18 = 3√2 | √20 = 2√5 | √24 = 2√6 |
|---|---|---|---|---|
| √27 = 3√3 | √28 = 2√7 | √32 = 4√2 | √40 = 2√10 | √44 = 2√11 |
| √45 = 3√5 | √48 = 4√3 | √50 = 5√2 | √52 = 2√13 | √54 = 3√6 |
| √56 = 2√14 | √60 = 2√15 | √63 = 3√7 | √68 = 2√17 | √72 = 6√2 |
| √75 = 5√3 | √76 = 2√19 | √80 = 4√5 | √84 = 2√21 | √88 = 2√22 |
| √90 = 3√10 | √92 = 2√23 | √96 = 4√6 | √98 = 7√2 | √99 = 3√11 |
Perfect squares and cubes to remember
Squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cubes: 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Frequently asked questions
How do you simplify a square root?
Find the largest perfect square that divides the number, then take its root outside the radical. For √72: 72 = 36 × 2, so √72 = √36 × √2 = 6√2.
What is simplest radical form?
A radical is in simplest form when the number under the root has no perfect-square factor other than 1 (or no perfect cube factor for a cube root), there is no fraction under the radical and no radical in a denominator.
How do you simplify a cube root?
Look for perfect cube factors such as 8, 27, 64 or 125. For example ∛54 = ∛(27 × 2) = 3∛2. With prime factors, every group of three identical primes comes out as one prime.
Can you simplify the square root of a negative number?
Not as a real number. √(−48) = √48 × √(−1) = 4√3 i, where i is the imaginary unit. Odd roots of negative numbers are real: ∛(−54) = −3∛2.
Why use prime factorization to simplify radicals?
Prime factors make every perfect power easy to spot. Write the number as a product of primes, then each group of k identical primes (k being the index) moves outside the radical as a single prime.
Is √50 the same as 5√2?
Yes. 50 = 25 × 2 and √25 = 5, so √50 = 5√2 ≈ 7.0711. Both forms have the same value; 5√2 is the simplified form.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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