How to Simplify a Radical Expression
A radical ⁿ√N is in simplest form when the radicand N contains no factor that is a perfect nth power (other than 1). Simplifying works by factoring the radicand into primes, then pulling any factor that appears in a complete group of n identical primes outside the radical, using the rule ⁿ√(aⁿ·b) = a·ⁿ√b. This calculator factors your radicand, applies that rule for the root index you choose, and multiplies the extracted factor by any coefficient you enter out front.
Formula and method
To simplify ⁿ√N by hand: (1) factor N into primes, e.g. 72 = 2³·3². (2) For each prime, divide its exponent by the index n and keep the quotient and remainder — for √72 (n = 2), 2³ gives one pair (2¹ outside) with 2¹ left inside, and 3² gives one pair (3¹ outside) with nothing left inside. (3) Multiply the outside factors together (2 × 3 = 6) and multiply the leftover inside factors together (2 remains), giving √72 = 6√2. The same process works for cube roots (n = 3), fourth roots (n = 4), and beyond — you're just grouping prime factors into sets of n instead of pairs.
Common mistakes
- Not finding the largest perfect power factor: √72 = √(4·18) = 2√18 is technically true but not fully simplified, since 18 still hides a perfect square (9). Always factor down to primes, or use the largest perfect nth-power divisor, to guarantee the result is fully reduced.
- Forgetting the index on higher-order roots: for a cube root, you need a factor cubed (e.g. 2³ = 8) to pull a term outside — a squared factor like 2² = 4 does not simplify ∛N, since it is not a perfect cube.
- Dropping the radical for a negative even-index root: √-16 has no real value; do not simplify it to -4 or 4. Only odd-index roots (cube root, fifth root, …) of negative numbers are real.
When simplified radical form matters
- Algebra and pre-calculus coursework typically requires answers in simplest radical form rather than as decimals, especially for the Pythagorean theorem, the quadratic formula, and distance/geometry problems.
- Simplified radicals make it easier to combine like terms — 3√2 + 5√2 = 8√2 is straightforward, while 3√2 + 5√8 hides that √8 = 2√2 until you simplify it.
- Engineering and physics formulas (e.g. RMS values, resonance frequencies) often carry irrational constants that are cleaner and more exact to report in radical form than as rounded decimals.