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Enter your values, then click Calculate result.How this calculator helps
This simplifying radical expressions solver handles a deliberately defined algebra task: sums and differences of square roots of non-negative integers, with integer coefficients and integer constants. It extracts square factors from each radicand and combines terms that share the same remaining square-free radicand. The output is an exact expression rather than a decimal approximation. Enter an expression such as 2sqrt(72)-sqrt(8)+3; square-root symbols are also supported for integer radicands. The parser does not execute the text as code, interpret variables or claim to be a general computer-algebra system.
How to use it
- 1
Enter up to twenty terms using integer coefficients, sqrt(integer) and plus or minus signs; an optional multiplication mark before sqrt is accepted.
- 2
Keep radicands between zero and 1,000,000,000, and original coefficients or constants within ±1,000,000.
- 3
Click Calculate result to extract square factors and collect matching radical terms.
- 4
Check the worked factorization independently. Editing text preserves the last submitted expression until Calculate is clicked again.
Formula and methodology
For n = q²r with r square-free, a√n = aq√r. Terms with the same r combine by adding their coefficients. A perfect-square radicand becomes an integer; √0 contributes zero.
The calculator applies the disclosed heuristic to values entered on this device. It does not load private admissions data or claim to reproduce an institution's review process.
Worked calculation example
2sqrt(72)-sqrt(8)+3 becomes 2×6√2 − 2√2 + 3 = 3 + 10√2. The exact root terms remain irrational even though the integer coefficient simplifies.
How to interpret your result
At most 20 integer square-root terms and constants, 1,000 characters, radicands ≤10⁹, original coefficients/constants ≤10⁶ in absolute value. No general CAS, variables, fractions or complex roots. Local deterministic parsing.
For different inputs or formulas, use Square Root Calculator; Scientific Notation Calculator; Fraction Calculator.
Related questions this calculator covers
- simplifying radical expressions solver
Scenario comparison
| Scenario | What it shows |
|---|---|
| 2sqrt(72)-sqrt(8)+3 = 3 + 10√2. | |
| sqrt(50)+sqrt(8) = 7√2. | |
| sqrt(9)-3 = 0. |
Common mistakes to avoid
- Adding radicands directly across separate roots.
- Combining roots with different square-free radicands.
- Entering an unsupported expression and assuming it is a general symbolic solver.
Factor each radicand into an extracted square and a square-free remainder, then collect coefficients. Optionally compare numerical sums with the separate square-root calculator as a secondary check.
Authoritative reference. OpenStax elementary algebra explains square-root properties and simplifying radicals; the bounded grammar and factorization method are disclosed here.What can affect the result?
Extract pairs of factors
The square-root identity uses pairs of identical prime factors to create an integer outside the radical. For seventy-two, the factorization is two cubed times three squared. One pair of twos and one pair of threes leave a single two under the root, so √72 becomes 6√2. The solver implements bounded trial factorization rather than guessing a nearby perfect square. Every paired factor moves outside; factors with odd multiplicity remain inside. Keeping the remaining radicand square-free makes like terms easier to recognize.
Combine like radicals after simplification
Two roots that look different at first may share the same simplified radical. √8 and √18 become 2√2 and 3√2, so their sum is 5√2. In contrast, √2 and √3 have different square-free radicands and cannot be combined into one coefficient times a single unchanged root. The calculator groups coefficients only after factor extraction. Exact cancellation can remove a radical entirely, and a result with no remaining terms displays zero rather than an empty expression.
Allowed syntax keeps the calculation predictable
Use forms such as sqrt(12), 3sqrt(12), 3*sqrt(12), -sqrt(12) or an integer constant. Terms may be joined by plus and minus signs, and whitespace is ignored. The square-root symbol can precede an integer or a parenthesized integer. The supported grammar excludes decimal coefficients, fractions, products of two roots, division, exponents, variables, nested roots and negative radicands. Those expressions can require different algebra or a complex-number branch. Unsupported syntax is rejected rather than partially parsed into an apparently valid answer.
Exact roots and decimal calculations answer different questions
The separate square-root calculator is appropriate when you want a numerical root of a supplied value. This page retains an exact radical sum and does not choose a decimal approximation as its primary output. For checking, you can approximate each original and simplified term independently and compare their sums, allowing for floating-point rounding. Numerical agreement is a useful cross-check but is not a complete symbolic proof. Factorization and coefficient collection explain why the expressions are algebraically equivalent in the supported real-valued domain.
Bounded expressions protect responsiveness
The text length, number of terms, radicand and coefficient limits prevent an unexpectedly expensive input from occupying the browser indefinitely. Trial factoring is bounded by the square root of each permitted integer and skips work once the remaining factor is small. The page does not call a remote algebra service or evaluate user text as JavaScript. Limits are visible because accepting arbitrary-length mathematical expressions would require a different implementation and verification strategy. For an unsupported school problem, rewrite it into a supported subexpression only if that preserves its mathematical meaning.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
At most 20 integer square-root terms and constants, 1,000 characters, radicands ≤10⁹, original coefficients/constants ≤10⁶ in absolute value. No general CAS, variables, fractions or complex roots. Local deterministic parsing.
Sources and review information
Frequently asked questions
Can this simplify √50 + √8?＋
Yes. The two terms become 5√2 and 2√2, and their sum is 7√2. Enter sqrt(50)+sqrt(8) or the equivalent square-root-symbol syntax. The solver displays exact coefficients rather than rounding the roots.
Does √a + √b equal √(a+b)?＋
Not in general. Addition must preserve separate roots unless factor extraction reveals matching square-free radicands. For example, √2 + √3 cannot be simplified to √5. This solver combines like radical terms, not radicands inside a sum.
Are negative numbers under a root supported?＋
No. This tool is limited to non-negative integer square roots in the real domain. A leading minus sign on a term is allowed, but sqrt(-4) would require complex-number handling and is rejected.
Can I enter variables or fractions?＋
No. Variable assumptions, rational coefficients and division require a broader algebra system. Supported expressions contain bounded integer constants, bounded integer coefficients and square roots of bounded non-negative integers joined by addition or subtraction.
Why does a perfect-square root become a constant?＋
If all prime factors occur in pairs, the remaining square-free radicand is one. For example, √36 becomes six, and six combines with other integer constants. A root of zero adds no term to the final expression.