Simplify Radicals Calculator

√72 and 6√2 are the same number written two different ways — but hand one to a teacher and they’ll mark the first one wrong. Simplest radical form isn’t about the value, it’s about the convention: no perfect square factors left inside the radical, no radicals in the denominator. The simplify radicals calculator above pulls out every extractable factor and returns the expression in the form examiners and textbooks expect.

What “Simplified” Actually Means

A radical expression is in simplest radical form when three conditions are met:

• The radicand has no perfect square factors (for square roots), no perfect cube factors (for cube roots), and so on.
• There are no fractions under the radical sign.
• There are no radicals in the denominator of a fraction.

√12 fails the first condition because 12 = 4 × 3, and 4 is a perfect square. √(1/5) fails the second. 1/√3 fails the third. The calculator handles all three cases and returns the expression in the form a√b, where b contains no further extractable factors.

The Simplification Process: Prime Factorization Method

The most reliable method — and the one that always works, regardless of how large the radicand is — is prime factorization. Break the radicand into its prime factors, group identical factors into pairs (for square roots), pull one factor out of each pair, and multiply everything outside the radical together.

√180, step by step:

1. Prime factorize: 180 = 2 × 2 × 3 × 3 × 5
2. Group into pairs: (2 × 2) × (3 × 3) × 5
3. Pull one from each pair: 2 × 3 = 6 outside the radical
4. Leave unpaired factors inside: √5 remains
5. Result: 6√5

Verification: (6√5)² = 36 × 5 = 180 ✓

The shortcut for smaller numbers — finding the largest perfect square factor directly — is faster but depends on recognizing that factor. Prime factorization requires no pattern recognition and works on any number.

Common Simplifications Reference

Simplifying Cube Roots

The process is identical but the grouping changes: instead of pairs, you look for triplets of identical prime factors. Pull one factor out per complete group of three; anything not forming a full triplet stays inside the radical.

∛54: prime factorize → 2 × 3 × 3 × 3. One triplet of 3s. Pull 3 outside, leave 2 inside. Result: 3∛2.

∛128: 128 = 2⁷ = 2³ × 2³ × 2. Two complete triplets of 2s. Pull 2 × 2 = 4 outside, leave 2 inside. Result: 4∛2.

Adding and Subtracting Radical Expressions

Like radicals — those with the same index and the same radicand — combine exactly like like terms. 3√2 + 5√2 = 8√2. Unlike radicals cannot be combined: √2 + √3 stays as √2 + √3, with no further simplification possible.

The catch: two radicals that look unlike may actually be like after simplification. 5√2 − √18 looks like it can’t combine — but √18 = 3√2, so the expression becomes 5√2 − 3√2 = 2√2. Always simplify each radical individually before deciding whether terms can be combined.

Rationalizing the Denominator

A fraction with a radical in the denominator — like 5/√3 — is not in simplest radical form. To rationalize it, multiply numerator and denominator by the same radical:

5/√3 × √3/√3 = 5√3/3

When the denominator is a binomial containing a radical — like 1/(2 + √5) — multiply by its conjugate (2 − √5):

1/(2 + √5) × (2 − √5)/(2 − √5) = (2 − √5)/(4 − 5) = (2 − √5)/(−1) = √5 − 2

The conjugate method works because (a + √b)(a − √b) = a² − b, which eliminates the radical from the denominator entirely.

The Product and Quotient Properties

Two properties govern nearly all radical simplification work:

• Product property: √(a × b) = √a × √b — allows splitting a radical into factors, which is the basis of the entire simplification process.
• Quotient property: √(a/b) = √a / √b — allows simplifying radicals that contain fractions by treating numerator and denominator separately.

What these properties do not allow: √(a + b) ≠ √a + √b. This is one of the most common algebra errors. √(9 + 16) = √25 = 5, not √9 + √16 = 3 + 4 = 7. The product and quotient properties apply to multiplication and division under the radical — never to addition or subtraction.