Exact simplified radicals first, decimals second, equation roots verified
A radical answer has two honest forms: the exact one (√180 is 6√5, and nothing else is quite true) and the decimal that approximates it. This calculator leads with the exact form: enter a coefficient, a root index and a radicand and it will simplify by extracting every perfect nth-power factor, then give the decimal. A second tool adds, subtracts, multiplies and divides two radical terms by simplifying each first, which is where like terms reveal themselves. A third solves simple radical equations the way they must be solved: isolate, raise to the power, and then check every candidate in the original equation, because squaring manufactures extraneous roots and only the check catches them.
Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Exact simplification is offered for integer radicands within the stated range; decimals are approximations, rounded for display.
What this calculator does
Radicals are where school mathematics starts insisting on exactness, and this page is built to that standard: simplification is done by prime factorisation, pulling each group of n identical factors out as one, which is the method itself rather than a search through square numbers. The combine tool applies the same simplification to both terms before operating, which is why 3√8 and 5√2 turn out to be combinable (the first is secretly 6√2). The equation solver follows the textbook protocol end to end and prints its verification step, because the rejected candidate is not an error in the working: it is the predictable artifact of squaring, and a solver that hides it teaches the wrong lesson.
How to use it
- Index defaults are your responsibility. The index is an input everywhere, since √ and ∛ are different questions; cube and higher roots of negative numbers are real for odd indices and are handled on the simplify tool, while even roots of negatives report the principal complex root.
- Coefficients stay outside until they cannot. In the combine tool, multiplication and division work on the whole terms (coefficients multiply, radicands multiply), and division rationalises the denominator, moving the radical upstairs per the standard convention.
- Decimals are the check, not the answer. Each exact result carries its decimal so you can sanity-check magnitude; where the two disagree with a classmate’s answer, the exact forms are what should be compared.
- Equation templates are the common shapes. One radical, isolated, equal to a constant or to x itself: most textbook and exam equations reduce to these after a line of algebra, and the verification discipline transfers unchanged to the harder ones.
The method
- Simplification: factor the radicand into primes. For each prime with exponent e, the factor contributes p^(e div n) outside the radical and p^(e mod n) remains inside. The result: coefficient × (outside part) × ⁿ√(inside part), exact when the radicand is an integer.
- Decimal value: coefficient × radicand^(1/n), with the real odd-root convention for negatives and the principal complex root a + bi for even indices of negatives.
- Combining: like radicals (same index, same simplified radicand) add and subtract by coefficient; multiplication multiplies coefficients and radicands then re-simplifies; division divides and rationalises by multiplying top and bottom to clear the denominator’s radical.
- Equations: square both sides, solve the resulting polynomial (linear for the constant templates, quadratic for √(ax+b) = x), then substitute every candidate into the ORIGINAL equation and keep only those that satisfy it.
Worked examples
√180. Factor: 180 = 2² × 3² × 5. Pairs come out: 2 × 3 = 6 outside, 5 stays inside: 6√5, about 13.4164. The same machinery on ∛72 (72 = 2³ × 3²) gives 2∛9, about 4.1602: one method, every index.
3√8 + 5√2. √8 simplifies to 2√2, so the first term is 6√2, and the sum is 11√2, about 15.5563. Terms that look unrelated combine exactly when their simplified radicands match, and not before: 6√2 + 5√3 stays stubbornly itself, which the tool reports rather than forcing.
√(2x + 3) = x. Squaring gives 2x + 3 = x², that is x² − 2x − 3 = 0, candidates x = 3 and x = −1. Checking in the original: √(9) = 3 ✓; √1 = 1 ≠ −1 ✗. Only x = 3 solves it, and the rejected candidate is the lesson: squaring cannot tell a root of the equation from a root of its negative twin.
Reading your results
Read the exact form as the answer and the decimal as its portrait: 6√5 is the value, 13.4164 is what it looks like on a number line. A simplified inside part of 1 means the radical vanished (the radicand was a perfect power), and the tool says so by printing a plain number. In the equation results, read the verification column as part of the solution, not a footnote: an unverified candidate is a hypothesis, and the difference between the two lists is precisely the extraneous-root phenomenon your course is testing.
Limitations
- Exact factorisation is for integers of moderate size. Radicands up to a million factor instantly here; beyond that, or for decimals and fractions, the decimal value is the result and the exact form is reported as unsimplified rather than faked.
- Nested radicals and sums under one radical are out of scope. Expressions like √(5 + 2√6) have a denesting theory of their own and are not simplified here; the tools take one radical per term, as stated.
- The equation solver covers its three templates. Equations with two radicals, radicals on both sides, or higher indices need additional isolation rounds; the protocol (isolate, power, verify) is the same, the algebra longer, and no verification step is ever optional.
- Complex output is the principal root. An even root of a negative has n complex values arranged around a circle; the page reports the principal one and says so, since the full set is a topic, not a line of output.
Frequently asked questions
How do you simplify a radical?
Factor the radicand and extract every group of n identical factors as a single factor outside: √180 is √(2² · 3² · 5), the pairs 2 and 3 come out, and 6√5 remains. If nothing extracts, the radical was already simple. The worked examples run the method at index 2 and 3 because the procedure is one idea at every index: groups of n, not pairs, once the index is general.
Why do some radical equations have extraneous solutions?
Because squaring erases sign: a² = b² whenever a = b OR a = −b, so the squared equation quietly unions the original with its negated twin, and the twin’s roots arrive as candidates. They are not mistakes and cannot be prevented by care in the algebra; they are filtered by the one step that knows the difference, substitution into the original equation, which is why this page’s solver prints the check for every candidate it finds.
Can you add radicals with different radicands?
Not directly, and not after simplification either, unless the radicands become equal in simplifying (as √8 became √2 in the worked example). √2 + √3 is a final answer, not a problem awaiting effort; what you can always do is approximate the sum decimally, which trades exactness for a number. The combine tool enforces the rule rather than papering over it, since accepting a forced combination is how √2 + √3 = √5 myths are born.
What is the cube root of a negative number?
A real negative number: ∛(−8) = −2, since (−2)³ = −8. Odd indices keep the sign because an odd power preserves it; even indices cannot (a square is never negative among the reals), which is why even roots of negatives leave for the complex numbers. The simplify tool follows exactly this convention and labels the even-negative case as the principal complex root.
Related calculators
Method: simplification by prime factorisation with nth-power extraction, combination after simplification with rationalised division, and equation solving by isolation, powering and mandatory verification of every candidate in the original equation. Last reviewed: 15 October 2026.