Radical Simplifier — Simplify Square Roots & nth Roots (Free)

A radical simplifier rewrites a root so no perfect power is left inside. To simplify √72, find the largest perfect square factor, 36, and split the root: √72 = √36 · √2 = 6√2 ≈ 8.4853. Cube and higher roots work the same way with perfect cubes or nth powers: ∛54 = ∛27 · ∛2 = 3∛2.

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How to use the radical simplifier

1. Pick a tab: Simplify a radical, Add & subtract, Multiply or Divide & rationalize.
2. To simplify one root, enter an optional coefficient, choose the index (2 for square roots, 3 for cube roots, up to 10) and type the radicand. It can be a whole number, a fraction, a decimal or a term with variables such as 18x^4y^3.
3. In the other tabs, type radicals with √, ∛, ∜ or the symbol buttons. You can also write sqrt(50), cbrt(16) or root(5, 32).
4. Read the simplest radical form, the decimal value and the grouping picture, then open Show steps for the full working.

How to simplify a radical

The rule behind every step is the product rule for radicals: ⁿ√(ab) = ⁿ√a · ⁿ√b, for a and b that are not negative. You split the radicand into a perfect power times whatever is left, then take the root of the perfect power.

1. Write the radicand as a product of primes. For 72, that is 2³ · 3².
2. Circle groups of equal factors, with as many in each group as the index: pairs for square roots, triples for cube roots.
3. Each full group leaves the root as one factor. Leftover factors stay inside.
4. Multiply what came out by any coefficient already in front.

For √72, the pairs are 2·2 and 3·3, with one 2 left over. So 2 · 3 = 6 comes out and 2 stays in: √72 = 6√2. For ∜162 = ∜(2 · 3⁴), the four 3s form one group, so ∜162 = 3∜2.

Radicals with variables

Divide each exponent by the index. The quotient is the power that comes out, and the remainder stays inside. For √(18x⁴y³): 18 = 9 · 2, x⁴ gives 4 ÷ 2 = 2 remainder 0, and y³ gives 3 ÷ 2 = 1 remainder 1. So √(18x⁴y³) = 3x²y√(2y). The calculator assumes the variables are positive, which is the usual convention in algebra classes, so no absolute value signs are needed.

Reducing the index

If every exponent inside shares a factor with the index, you can divide both by it. ⁶√8 = ⁶√(2³) = 2^(3/6) = 2^(1/2), which is √2. Many tools stop at ⁶√8; the calculator reduces it for you.

Perfect squares, cubes and worked examples

Knowing the first few perfect powers makes the search quick. Perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Perfect cubes: 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Adding and subtracting radicals

Only like radicals combine: same index and same radicand. Treat them like like terms in algebra. Always simplify first, because radicals that look different often match after simplifying.

Example: 3√12 + 5√27 − √48. Simplify each one: √12 = 2√3, √27 = 3√3 and √48 = 4√3. The expression becomes 6√3 + 15√3 − 4√3 = (6 + 15 − 4)√3 = 17√3 ≈ 29.4449.

Multiplying radicals

With the same index, multiply the outside numbers together and the radicands together, then simplify: 2√6 · 3√15 = 6√90 = 6 · 3√10 = 18√10. For binomials, distribute (FOIL) and combine like terms. Conjugates such as (2 + √3)(2 − √3) give a whole number, because (a + b)(a − b) = a² − b² = 4 − 3 = 1.

Different indexes need a common index first: √2 · ∛2 = ⁶√(2³) · ⁶√(2²) = ⁶√32 ≈ 1.7818.

Dividing radicals and rationalizing the denominator

Simplest radical form has no root in a denominator. To clear one:

• One square root: multiply top and bottom by that root. 3/√2 = 3√2/(√2 · √2) = 3√2/2.
• A cube or higher root: multiply by whatever makes the radicand a perfect power. 5/∛4: since 4 · 2 = 8 = 2³, multiply by ∛2 to get 5∛2/2.
• Two terms: multiply by the conjugate. 6/(3 − √5) = 6(3 + √5)/(9 − 5) = (18 + 6√5)/4 = (9 + 3√5)/2 ≈ 7.8541.

Common mistakes when simplifying radicals

Where simplified radicals show up

Exact radicals appear in the Pythagorean theorem (a square with side 1 has diagonal √2), in special right triangles (30-60-90 sides 1, √3, 2), in the quadratic formula, in distance and midpoint problems, and in geometry answers that tests expect in exact form rather than as rounded decimals.

Simplifying radicals practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is simplified radical form?

A radical is in simplest form when no perfect power factor is left under the root, there is no fraction under the root, and no radical is left in a denominator. The index should also be as small as possible, so ⁶√8 becomes √2.

What is radical 12 simplified?

√12 = √4 · √3 = 2√3, about 3.4641. Check it by squaring: (2√3)² = 4 · 3 = 12.

What is radical 32 simplified?

√32 = √16 · √2 = 4√2, about 5.6569. Stopping at 2√8 is a common slip, because 8 still contains the perfect square 4.

How do you simplify radical expressions with variables?

Simplify the number and each variable separately. Divide each exponent by the index: the quotient comes out and the remainder stays in. For example, √(45x⁵) = 3x²√(5x).

How do you simplify radical fractions?

Take the root of the top and bottom separately, then rationalize. √(5/9) = √5/3. If the denominator is not a perfect square, multiply inside first: √(5/8) = √(10/16) = √10/4.

Can a TI-84 simplify radicals?

Standard TI-84 models show a decimal (√72 gives about 8.485), not the exact form 6√2. Use the calculator above for the exact answer and steps, and the decimal to check it.

Related calculators

• Square root simplifier for quick square roots of whole numbers
• Cube root calculator for cube roots as decimals and exact forms
• Prime factorization calculator
• Exponent rules calculator for rational exponents
• FOIL calculator
• Special right triangles calculator