Starlight Tools

Radical Expression Simplifier with Steps

Reduce numeric square roots, cube roots, and nth roots using exact arithmetic. Your expression stays in this browser.

Enter one radical expression to extract perfect powers, apply rational coefficients, combine like radicals, and see a decimal check.

Enter a radical expression

Use sqrt(72), cbrt(54), root(4,48), or symbols such as √72. Rational coefficients and divisors are allowed.

Input rules and limits

Enter a sum or difference of rational numbers and single radical terms. A term may have a coefficient before the radical or a rational multiplier/divisor after it, such as (3/4)sqrt(32) without the outer parentheses: 3/4sqrt(32). Root indices must be 2–12 and radicands must be integers from −1,000,000,000,000 to 1,000,000,000,000. Products of separate radicals, nested radicals, variables, and radical denominators are not supported.

Try an example

Simplified result

Exact simplified expression
—
Enter an expression to begin.
Terms read—
Radicals reduced—
Like terms—
Approximation—

Step-by-step simplification

Perfect-power extraction and like-term steps will appear here.

How to simplify radical expressions

1. Factor the radicand

Write the number under the radical as prime factors. The index tells you how large each complete group must be.

2. Extract perfect powers

For an nth root, move one factor outside for every group of n identical factors inside.

3. Multiply coefficients

Multiply extracted factors by the coefficient already attached to the radical. Fractions remain exact.

4. Combine like radicals

Add or subtract coefficients only when both the root index and reduced radicand are the same.

Core rule

root(n, anb) = a·root(n, b) for a ≥ 0.

For example, √72 = √(6²·2) = 6√2. Then 6√2 + 4√2 − 3√2 = 7√2 because all three terms have the same square-root index and radicand.

Worked examples

Square root

√180 = √(6²·5) = 6√5. The remaining radicand 5 has no square factor greater than 1.

Cube root

∛54 = ∛(3³·2) = 3∛2. A group of three 3s moves outside a cube root.

Fourth root

∜48 = ∜(2⁴·3) = 2∜3. Four equal factors of 2 form one factor outside.

Negative odd root

∛(−250) = −∛250 = −5∛2. Negative radicands remain real for odd indices.

Calculation method and limits

Integer radicands are prime-factorized with exact BigInt arithmetic. Exponents are divided by the root index to identify the outside and inside factors. Coefficients use reduced BigInt fractions, so the exact result is not based on floating-point rounding. The decimal value is shown separately as a numerical check.

The calculator works over the real numbers. Even roots require nonnegative radicands; odd roots accept negative radicands. It simplifies sums and differences of single numeric radical terms with indices 2–12, but not variables, nested radicals, radical denominators, or products of separate radicals. Last reviewed: 4 August 2026.

Radical simplification FAQs

How do you simplify a radical?

Factor the radicand, group factors in sets equal to the root index, move one factor from each complete group outside, and leave ungrouped factors inside.

When can radical terms be combined?

They are like terms only when their root indices and simplified radicands match. For example, 2√3 + 5√3 = 7√3, but √3 + ∛3 cannot be combined.

Can this tool simplify negative radicands?

Yes, for odd root indices. The cube root of −8 is −2. Even roots of negative numbers are not real, so the tool reports a clear error for those inputs.

What radical syntax can I enter?

Use sqrt(n), cbrt(n), root(index,n), √n, ∛n, or ∜n. Coefficients can be integers, finite decimals, or fractions.

Does the simplifier use decimal rounding?

Not for the exact answer. Coefficients are reduced fractions and radicands are integers. Only the separately labeled approximation uses decimal rounding.

Does this tool rationalize radical denominators?

No. Rational divisors such as sqrt(8)/3 are supported, but a radical denominator or product of separate radical factors is outside this calculator’s stated input grammar.

Explore more tools