Simplify square-root expressions with exact answers
This simplifying radicals calculator extracts perfect-square factors, combines like square roots and performs exact addition, subtraction, multiplication and division of supported real numeric radical expressions. It can rationalize a denominator containing square-root terms, including common conjugate examples such as 1/(√2 + 1).
The exact radical form is the main answer. A separate approximate decimal helps with a numerical comparison but does not replace the exact expression. For example, √72 becomes 6√2 rather than only a rounded decimal. This is a different task from simply evaluating a principal square root with the Square Root Calculator.
This version handles square-root expressions with numeric radicands, not every possible radical problem. Cube roots, symbolic variables, complex values and nested irrational radicands are outside the supported scope. Those restrictions are explicit so that a numeric square-root simplification is not mistaken for a general computer algebra system.
How to use the radical simplifier
- Choose Simplify for one expression, or Add, Subtract, Multiply or Divide for a pair of expressions.
- Use sqrt(...) for a square root. You can also enter √72 or √(72). For a fractional radicand, use sqrt(3/4), not √3/4 unless that division outside the root is intentional.
- Enter parentheses around a full denominator, such as 1/(sqrt(2) + 1). Use * for explicit multiplication or write 2sqrt(3).
- Calculate and compare the exact result with the value-specific working and approximate decimal panel.
Editing an expression clears its old result. The second field is disabled in Simplify mode and becomes available for two-expression operations. Reset returns to the blank single-expression form. Copy, Download and Print keep the original inputs, exact result and calculation steps.
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For an optional numerical check, browse scientific calculators (paid link). Confirm the supported functions; a numerical device does not necessarily reproduce this tool’s symbolic working. No purchase is needed.
Extract square factors and combine like radicals
For a nonnegative integer radicand, separate the largest perfect-square factor. Since 72 = 36 × 2, √72 = 6√2. For a rational radicand a/b with b positive, the calculator uses √(a/b) = √(ab)/b, then extracts square factors and reduces the rational coefficient.
After simplification, only matching radicands can be combined by adding coefficients. The expressions √72 and √8 become 6√2 and 2√2, so their sum is 8√2. In contrast, √2 + √3 cannot be replaced by √5. Multiplication is different: √2 × √8 = √16 = 4 for these nonnegative radicands.
All coefficients are exact reduced fractions. Ordinary decimals are converted into rational numbers before root extraction. The tool uses principal nonnegative square roots; a minus sign outside a root changes the coefficient, not the meaning of the principal root.
Rationalizing a radical denominator
A square root in a denominator can often be removed by multiplying by an appropriate expression equal to one. For 1/√3, multiply top and bottom by √3 to obtain √3/3. The displayed coefficient form is (1/3)√3.
For 1/(√2 + 1), multiply by the conjugate (√2 − 1)/(√2 − 1). The denominator becomes 2 − 1 = 1, leaving √2 − 1. The calculator may display this as −1 + √2 because it orders the rational term first.
For more involved supported denominators, the exact arithmetic can apply successive conjugations. This is subject to the stated six-prime and term limits; it is not an unrestricted nested-radical solver. A denominator that simplifies to zero is always rejected, even if its unsimplified form looked nonzero.
Worked radical simplification examples
Extracting a square: Enter sqrt(72) in Simplify mode. The result is 6√2 because √36 = 6 and the remaining radicand 2 has no square factor greater than one.
Combining like radicals: Enter sqrt(72) + sqrt(8), or use Add with those expressions in separate fields. The result is 8√2. The decimal check is approximately 11.313708499, while the exact answer remains 8√2.
A fractional radicand: sqrt(3/4) becomes (1/2)√3. The Fraction Calculator can help check the rational coefficient arithmetic independently.
Division: Divide sqrt(2) by sqrt(3). Rationalization gives (1/3)√6. For 1/(sqrt(2) + 1), the answer is −1 + √2. Squaring or multiplying a result back can be a useful exact check when the original divisor is nonzero.
Supported roots, precision and safety limits
Inputs may contain ordinary numeric literals, parentheses, +, −, multiplication, division and integer literal powers from −6 through 6. Root syntax is sqrt(...), √number or √(...). The contents of sqrt must reduce to a nonnegative rational number. For example, sqrt(2 + 2) is supported, but sqrt(2 + sqrt(2)) is not.
Each input has a 240-character, 160-token and 20-level nesting limit. Numeric literals have magnitude at most 1,000,000 and no more than six decimal places. For exact root extraction, the reduced radicand numerator multiplied by its positive denominator must not exceed 1,000,000,000,000. This complexity limit can reject a large fraction even when its decimal value looks small.
A calculation can introduce at most six distinct square-free prime factors and produce at most 32 expanded radical terms. Exact coefficients have a 256-digit intermediate safety limit. Variables, negative radicands, complex values, cube roots, equations and arbitrary functions are not accepted. A zero denominator and literal 0^0 are rejected.
The decimal panel uses floating-point arithmetic and may lose relative accuracy when large terms nearly cancel. The exact radical form is authoritative within the supported operations. Use the Scientific Calculator for an optional separate numerical evaluation, and do not treat this educational output as a measurement with unlimited precision.
Simplifying Radicals Calculator FAQs
Sources and methodology
These educational references explain the underlying mathematics. This independently implemented tool is not affiliated with their publisher. Supported methods, numerical precision and restrictions are described above.