Examples · Basic Math
Nth Root Examples: 8 Worked Problems
Eight roots, ordered so each one adds exactly one new idea to the last — every simplification here is checked by raising it back to the original number.
Perfect powers, solved in one line
The first three examples are exact whole-number roots — nothing to split, nothing left over.
- ⁵√3232 = 2⁵
- 2the fifth root of 2⁵ is 2, exactly
- ⁴√625625 = 5⁴
- 5the fourth root of 5⁴ is 5, exactly
- √400400 = 20²
- 20the square root of 20² is 20, exactly
Recognising these on sight comes from knowing a handful of perfect powers by heart — the same shortcut that makes factoring quadratics faster once the common squares are familiar.
Partial simplification: something comes out, something stays
- ∛5454 = 27 · 2, and 27 is a perfect cube
- ∛(27 · 2)split the radicand
- 3∛2∛27 = 3, and 2 has no cube factor left
- √1212 = 4 · 3, and 4 is a perfect square
- √(4 · 3)split the radicand
- 2√3√4 = 2, and 3 has no square factor left
- ∛200200 = 8 · 25, and 8 is a perfect cube
- ∛(8 · 25)split the radicand
- 2∛25∛8 = 2, and 25 has no cube factor left — 25 is a perfect square, not a perfect cube
Negative radicands: real for odd roots, undefined for even
- ∛−125−125 = (−5)³
- −5the cube root of a negative perfect cube is negative, exactly
A parallel fourth-root example would decline entirely — ⁴√−125 has no real value, since no real number raised to an even power is negative. The sign of the radicand and the parity of the index have to be checked before anything else.
A large radicand, and a root that stays irrational
- ⁶√1,000,0001,000,000 = 10⁶
- 10the sixth root of 10⁶ is 10, exactly
Not every large number is this tidy. ¹⁰√2 has no perfect tenth-power factor at all — it stays as an irrational radical, approximately 1.072, and no amount of searching will simplify it further. A big radicand is not, on its own, a hint that a root will simplify; what matters is whether it hides a matching perfect power.
Checking every result the same way
Raise the simplified answer back to the original index and confirm it reproduces the radicand.
| Example | Simplified | Check |
|---|---|---|
| 4 | 3∛2 | 3³ · 2 = 27 · 2 = 54 ✓ |
| 5 | 2√3 | 2² · 3 = 4 · 3 = 12 ✓ |
| 6 | 2∛25 | 2³ · 25 = 8 · 25 = 200 ✓ |
This is exactly the check the Nth Root Calculator on this site runs internally — the perfect-power factor it pulls out is verified, by construction, to multiply back to the original number exactly.
Questions about nth root examples
What is the cube root of 54?
3∛2. 54 splits into 27 · 2, and 27 is a perfect cube — its cube root, 3, comes out front, leaving 2 with no cube factor left inside.
What is the cube root of a negative perfect cube?
Negative and exact. ∛−125 = −5, because (−5)³ = −125. This only works for an odd index; an even root of a negative number has no real value.
Why doesn't 25 simplify out of ∛200?
Because 25 is a perfect square, not a perfect cube, and the root being simplified here is a cube root. Only a factor matching the root's own index can be pulled out.
How do I check a simplified root is correct?
Raise it back to the original index. For 3∛2, check 3³ · 2 = 27 · 2 = 54 — if it does not reproduce the starting number, a wrong factor was pulled out.