Simplify Radicals Calculator

Simplify any radical expression instantly. Enter a radicand and root index to extract the integer coefficient, reduced radicand, or decimal equivalent.

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Inputs

Radicand (number under the radical)

Radical Index (n in nth root)

What to Display

Simplified Result

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Get a plain-English breakdown of your result with practical next steps.

Simplified Result—

The formula

How the
result is
computed.

How the Simplify Radicals Calculator Works

The simplify radicals calculator applies a systematic prime-factorization algorithm to reduce any radical of the form ⁿ√x into its fully simplified equivalent: k · ⁿ√m. Here, k is the integer coefficient extracted outside the radical sign and m is the reduced radicand — a positive integer retaining no perfect nth-power factors. According to Paul's Online Math Notes (Lamar University), a radical expression is considered fully simplified when the radicand contains no factors that are perfect nth powers, making further extraction impossible.

The Core Formula

The governing identity is: ⁿ√x = k · ⁿ√m, where x = kⁿ · m and every prime in m appears with an exponent strictly less than n. The coefficient k accumulates one copy of each prime base for every complete group of n identical prime factors found in x. The reduced radicand m holds the leftover primes that did not complete a full group. When m = 1, the original radicand was a perfect nth power and the result is a whole integer. When k = 1, no extraction is possible and the radical is already irreducible.

Understanding the Variables

Radicand (x): The positive integer placed under the radical sign. In √48, the radicand is 48. This value drives the entire factorization process.
Radical Index (n): The root being taken. Index 2 denotes a square root, 3 a cube root, 4 a fourth root, and 5 a fifth root. Convention assumes n = 2 when no index is written.
Coefficient (k): The integer multiplied outside the simplified radical. It equals the product of each prime base raised to the floor of (its exponent in x) divided by n.
Reduced Radicand (m): The integer remaining inside the radical after all complete nth-power groups have been extracted. Every prime factor in m appears with an exponent in the range 1 through n − 1.

Step-by-Step Simplification Method

Prime factorize the radicand. For example, 72 = 2³ · 3².
Divide each prime exponent by the index n using integer (floor) division to find the quotient q and the remainder r.
Compute k as the product of each prime base raised to its quotient. For √72 (n = 2): prime 2 has exponent 3 ÷ 2 = quotient 1 remainder 1, contributing 2¹ = 2; prime 3 has exponent 2 ÷ 2 = quotient 1 remainder 0, contributing 3¹ = 3. So k = 2 · 3 = 6.
Compute m as the product of each prime base raised to its remainder. For √72, only prime 2 has remainder 1, so m = 2¹ = 2.
Write the result as k · ⁿ√m. For √72 the simplified form is 6√2, with decimal value ≈ 8.485.

Worked Examples

Square Root (n = 2) — Simplify √200: Factorize: 200 = 2³ · 5². Quotients: 2→1, 5→1. Remainders: 2→1, 5→0. Therefore k = 2 · 5 = 10 and m = 2. Result: 10√2 ≈ 14.142.

Cube Root (n = 3) — Simplify ∛432: Factorize: 432 = 2⁴ · 3³. Quotients (÷3): 2→1, 3→1. Remainders: 2→1, 3→0. Therefore k = 2 · 3 = 6 and m = 2. Result: 6∛2 ≈ 7.560.

Already Irreducible — Simplify √7: Since 7 is prime with exponent 1 and 1 ÷ 2 yields quotient 0, no extraction occurs. k = 1 and m = 7. The radical √7 is already fully simplified.

Real-World Applications

Simplified radical forms appear throughout STEM disciplines. Geometry relies on them when computing diagonal lengths via the Pythagorean theorem — for example, a square with side 5 has a diagonal of 5√2, not the unwieldy √50. Physics uses simplified radicals in wave-speed formulas and orbital-period calculations. West Texas A&M University's Virtual Math Lab notes that working in simplified form reduces arithmetic errors when adding or subtracting like radical terms and makes verifying algebraic identities substantially easier. Standardized tests including the SAT and ACT regularly require students to recognize and produce simplified radical expressions as a core algebra competency.

Reference

Frequently asked questions

What does it mean to simplify a radical expression?

Simplifying a radical means rewriting ⁿ√x as k · ⁿ√m, where k is the largest integer coefficient extractable outside the radical and m is the smallest possible radicand. For example, √50 simplifies to 5√2 because 50 = 25 × 2 and √25 = 5. The expression is fully simplified when no perfect nth-power factors remain inside the radical sign.

How do you simplify a square root step by step?

First, find the prime factorization of the radicand. Then group the prime factors into pairs (since n = 2 for square roots). Each complete pair sends one copy of that prime factor outside the radical. Any unpaired primes stay inside. For √180: 180 = 2² · 3² · 5. The pairs yield coefficient k = 2 · 3 = 6, and the unpaired 5 remains inside, giving the simplified form 6√5.

What is the radical index and how does it change the result?

The radical index n tells the calculator which root to take. For n = 2 (square root), prime factors need to appear at least twice to exit the radical. For n = 3 (cube root), they must appear at least three times. For example, ∛54 simplifies to 3∛2 because 54 = 2 · 3³ and only the complete group of three 3s exits, giving k = 3 and m = 2. Changing the index changes which factor groups qualify, so the same radicand can simplify differently under different indices.

Can every radical be simplified to a whole number?

No. A radical produces a whole number only when the radicand is a perfect nth power — for instance, √144 = 12 and ∛27 = 3. When the radicand's prime factorization contains factors that do not form complete groups of n, an irrational radical term remains. The values √2, √3, √5, and √7 are all irrational: each has coefficient k = 1 and reduced radicand equal to the original value, meaning no simplification is possible.

What is the difference between the coefficient and the reduced radicand after simplification?

The coefficient k is the whole number placed outside the radical sign; it represents all the perfect nth-power content extracted from the original radicand. The reduced radicand m is the integer left inside the radical, retaining only prime factors whose exponents were too small to form a full group. In the expression 4√3, the coefficient is 4 and the reduced radicand is 3. Raising k to the nth power and multiplying by m always recovers the original radicand x.

Why does the calculator sometimes output 1 for the coefficient or the reduced radicand?

A coefficient of 1 signals that the original radicand contains no perfect nth-power factors, so nothing can be extracted — the radical is already in its simplest form. For example, √13 gives k = 1 because 13 is prime. A reduced radicand of 1 signals the opposite: the entire radicand was a perfect nth power and fully simplifies to a whole number. For instance, √49 yields k = 7 and m = 1, so the result is simply 7 with no radical symbol remaining.