Method · Basic Math
How to Simplify an Nth Root: Step by Step
Simplifying a root is finding the largest perfect power hiding inside the radicand and pulling it out — the same method for a square root, a cube root, or any root beyond.
The one method, for any index
Simplifying ⁿ√x always follows the same three moves, whatever n happens to be.
- Find the largest factor of x that is a perfect nth power.
- Split x into that factor times whatever remains.
- Pull the perfect-power factor out from under the root; leave the remainder inside.
The only thing that changes between a square root, a cube root and a tenth root is which list of perfect powers you are checking factors against.
Worked example: a cube root
- ∛24the expression
- 24 = 8 · 38 is the largest perfect cube dividing 24 — 2³ = 8
- ∛(8 · 3)split the radicand
- ∛8 · ∛3a root of a product is the product of the roots
- 2∛3∛8 = 2, since 2³ = 8
Worked example: a fourth root
- ⁴√48the expression
- 48 = 16 · 316 is the largest perfect fourth power dividing 48 — 2⁴ = 16
- ⁴√16 · ⁴√3split and separate
- 2⁴√3⁴√16 = 2, since 2⁴ = 16
The pattern is identical to the cube-root example — only the list of perfect powers being checked changed, from cubes to fourth powers.
Finding the largest perfect-power factor efficiently
Rather than guessing, check whole numbers in order — 2, 3, 4, … — raised to the root's index, and see which ones divide the radicand evenly.
- 2³ = 8does 8 divide 500? 500 ÷ 8 = 62.5 — no
- 3³ = 27does 27 divide 500? no
- 4³ = 64does 64 divide 500? no
- 5³ = 125does 125 divide 500? 500 ÷ 125 = 4 — yes
- ∛500 = ∛(125 · 4) = 5∛4125 is the largest perfect cube factor
For larger numbers, prime factorization is faster: break the radicand into primes, then group the exponents in sets matching the root index. 500 = 2² · 5³ has one complete group of three 5s, which is exactly the 125 found above.
When a root has nothing to simplify
∛10 has no perfect cube factor above 1 — 10 = 2 · 5, and neither prime appears three times. The root is already in simplest form, and no amount of searching will find a factor that isn't there. Reporting the decimal approximation, roughly 2.154, is a separate task from simplifying — it is not what 'simplest form' means.
Even roots of negative numbers, and odd roots that stay negative
Before simplifying, check the sign against the index. ⁴√−16 has no real value — no real number raised to an even power is negative. ∛−8 does have a real value, −2, since (−2)³ = −8; the simplification method works exactly the same way, just carrying the negative sign through to the answer.
- ∛−54the expression
- −54 = −27 · 227 is the largest perfect cube dividing 54
- ∛−27 · ∛2split the radicand, keeping the sign with the perfect-cube part
- −3∛2∛−27 = −3, since (−3)³ = −27
Checking your answer, always
Raise the simplified answer back to the nth power and confirm it reproduces the original radicand. For 2∛3, that means checking (2∛3)³ = 2³ · 3 = 8 · 3 = 24 — which matches the number the root started with.
The Nth Root Calculator on this site runs exactly this kind of check internally: the factor it pulls out is always verified, by construction, to multiply back to the original number exactly.
Questions about how to simplify an nth root
How do I simplify a cube root?
Find the largest perfect cube that divides the radicand, split the radicand into that factor times the remainder, then pull the perfect cube's cube root out. ∛24 = ∛(8·3) = 2∛3.
What if no perfect-power factor exists?
Then the root is already in simplest form — there is nothing to pull out. ∛10 cannot be simplified further; it is an irrational number, not an unfinished calculation.
Can you simplify the root of a negative number?
Only for an odd index. ∛−54 = −3∛2, following the same method as a positive radicand with the negative sign carried through. An even root of a negative number has no real value to simplify.
How do I check a simplified root is correct?
Raise it back to the original index. (2∛3)³ should equal 24 — 2³ times 3, which is 8 times 3, which is 24. If it doesn't match, a wrong factor was pulled out.