Root Calculator
Find and simplify square roots, cube roots, or any n-th root of a number. This calculator doesn't just give you a decimal — it shows the prime factorization, which factors pull outside the radical and which stay inside, and exactly where the answer sits on a number line.
Background
The n-th root of a number x is the value that, multiplied by itself n times, gives back x. Most roots of whole numbers are irrational — but many can still be written in a simpler exact form (like √72 = 6√2) by grouping the number's prime factors into sets of n and pulling every complete set outside the radical sign.
How to use this calculator
- Choose Square Root or Cube Root for the two most common cases, or General Root to set any whole-number index n.
- Type in your number and click Calculate. Negative numbers are allowed for odd indexes (like cube roots) but not for even ones.
- Whole numbers get a full prime-factorization breakdown showing exactly how the simplified form was built.
- If the answer is irrational, the number line shows exactly which two whole numbers it falls between.
How radical simplification works
Every whole number breaks down into a unique product of prime factors (the Fundamental Theorem of Arithmetic) — e.g. 72 = 2³ × 3².
To simplify an n-th root, group each prime's copies into sets of n. Every complete set of n identical primes can be pulled out of the radical as a single copy of that prime.
Leftover copies — fewer than n of them — can't form a complete group, so they stay trapped inside the radical.
Odd-index roots accept negative numbers, because a negative number raised to an odd power stays negative. Even-index roots need a nonnegative input, since any real number raised to an even power is nonnegative — so no real number squared (or to the 4th, 6th...) can equal a negative.
A number is a perfect n-th power exactly when every one of its prime exponents is a multiple of n — then every factor pulls outside and nothing is left under the radical.
Formula & Equations Used
Definition: ⁿ√x = x^(1/n)
Product rule: ⁿ√(a · b) = ⁿ√a · ⁿ√b
Simplification: if x = p₁^e₁ · p₂^e₂ · ..., then the part that leaves the radical is p₁^⌊e₁/n⌋ · p₂^⌊e₂/n⌋ · ... and the part that stays inside uses the remainder of each exponent, e mod n.
Domain rule: for even n, real results require x ≥ 0.
Example Problems & Step-by-Step Solutions
These work through a few cases the Quick Examples chips above don't cover — try the calculator itself for interactive step-by-steps on square roots, cube roots, perfect powers, and negative radicands.
Example 1 — An applied problem
A square garden has an area of 200 square feet. What's the exact side length?
Step: side length = √200. Factor 200 = 2³ × 5². Grouping into pairs: 2³ gives one pair plus a leftover 2; 5² is a complete pair.
Result: outside = 2 × 5 = 10, inside = 2 — the side length is exactly 10√2 feet (≈ 14.14 ft).
Example 2 — Already in simplest form
Simplify √15.
Step: 15 = 3 × 5. Neither prime appears more than once, so there's no complete pair to pull out from under the radical.
Result: nothing simplifies — √15 is already in simplest form (≈ 3.873, irrational).
Example 3 — A 4th root
Find the 4th root of 48.
Step: 48 = 2⁴ × 3. Grouping into sets of 4: 2⁴ is exactly one complete group; the single 3 has no group to join.
Result: outside = 2, inside = 3 — so the 4th root of 48 = 2⁴√3.
Example 4 — A leftover that isn't prime
Simplify ∛108.
Step: 108 = 2² × 3³. Grouping into triples: 3³ is a complete group and pulls out whole; the 2² has only 2 copies, short of a triple, so both stay inside together as 4.
Result: outside = 3, inside = 2² = 4 — so ∛108 = 3∛4 (the leftover under the radical doesn't have to be a single prime).
Frequently Asked Questions
Why can't I take an even root of a negative number?
Any real number raised to an even power comes out nonnegative — a negative times itself an even number of times always flips back to positive. So there's no real number that, squared or raised to the 4th, 6th, etc., gives a negative result.
What does "simplified radical form" mean?
It's the exact value written as a whole number times the smallest possible radical — e.g. 6√2 instead of the decimal 8.485... It keeps the answer exact rather than rounding it.
Is every non-perfect-power root irrational?
Yes, for whole-number inputs. If a whole number isn't a perfect n-th power, its n-th root is always irrational — it never terminates or repeats as a decimal.
Can decimals be put into exact radical form here?
Not with this tool's simplification step, since prime factorization only applies cleanly to whole numbers. Decimal radicands still get an accurate decimal approximation, just not a simplified exact form.
What's the difference between "evaluate" and "simplify"?
Evaluating gives a decimal approximation (like 8.485...). Simplifying keeps the answer exact by rewriting it with a smaller number under the radical (like 6√2). This calculator shows both side by side.
Why does the index matter for negative numbers?
It comes down to parity: an odd index preserves the sign of the input (so negative numbers have real odd roots), while an even index destroys sign information, which is exactly why even roots of negative numbers don't exist among the real numbers.
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