Student checking a perfect square
Know whether the result is exact.
Enter a nonnegative integer perfect square and inspect the exact-integer label and square check.
Reads √144 = 12 without unnecessary decimals or approximation signs.
Square Root Calculator
Find the nonnegative principal real square root, labeling exact integer roots separately from rounded approximations and equation solutions.
Use this result well
- Inputs that matter
- One nonnegative real radicand and a whole display precision from 0 through 12 decimal places
- Output to expect
- Nonnegative principal real root, exact-integer or rounded label, displayed-root square, residual and separate x²=n solutions
- How it works
- Applies the principal-value convention; integer perfect squares are exact, while other displayed roots are rounded to the requested precision and checked as rounded values
- √n denotes the nonnegative principal root; for n > 0 the equation x² = n has both +√n and −√n solutions.
- Negative radicands require complex-number methods outside this real-only tool, and a rounded approximation should not be treated as an exact identity.
Choose your path
Built around the job you need to finish
Return the nonnegative principal real square root while distinguishing exact integer roots, rounded approximations, and equation solutions.
Learner exploring an irrational root
Control visible rounding and understand residual error.
Enter a non-perfect square plus 0–12 decimal places.
Sees a rounded principal root and does not mistake the rounded square for an identity.
Solver working with x² = n
Distinguish radical notation from equation roots.
Read the principal-root result and the separate equation-solutions line.
Understands √n is nonnegative while x²=n has ±√n for n>0.
Authoritative checks for this tool
Outputs and checklists are planning aids. Review the linked current authorities and the records, terms, instructions, and requirements that apply to your exact situation before a consequential decision.
- DLMF §4.2 — Definitions of powers and principal valuesNational Institute of Standards and Technology · Official principal-value convention for powers and square roots.
- DLMF §4.30 — Elementary PropertiesNational Institute of Standards and Technology · Official real-function tables stating that square roots use real, nonnegative, finite principal values.
- Prealgebra 2e — Properties of IntegersOpenStax, Rice University · Educational distinction between principal square-root notation and positive/negative solutions when undoing a square.
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Reference & details
How it works
Updated August 2026
How it works
Updated August 2026Principal-value convention
For a nonnegative real radicand, the calculator returns the nonnegative principal root.
r = √n, where r ≥ 0 and r² = nExact integer roots
When a supported integer radicand is a perfect square, the integer root is shown without padded decimals or an approximation sign.
√144 = 12 exactlyRounded approximations
Non-integer displayed roots use the selected 0–12 decimal places. Squaring that displayed value and showing its residual makes the effect of rounding explicit.
Residual = (displayed root)² − radicandUpdated: August 2026
Example Scenarios
A student enters 144 and sees principal square root 12 labeled as an exact integer.
A learner enters 2, selects six decimal places, and sees 1.414214 as a rounded principal-root display with a nonzero residual.
A solver checks x² = 9 and sees the separate equation solutions ±3 while the principal square root remains √9 = 3.
Common Mistakes to Avoid
Common Mistakes to Avoid
Writing √9 = ±3
The principal square root is √9 = 3. The separate equation x² = 9 has the two solutions x = ±3.
Treating a rounded square check as exact
For a non-perfect square, the displayed root is rounded. Review the residual instead of claiming the rounded display squares to the radicand exactly.
FAQ
For a nonnegative real number n, √n denotes the nonnegative real number whose square is n.
When n is positive, both +√n and −√n square to n. The radical √n itself still denotes only the nonnegative principal root.
This tool labels safe integer perfect-square roots exact, such as √144 = 12. Other displayed values are treated as rounded approximations.
It is the displayed rounded root squared minus the original radicand. A small nonzero value comes from display rounding, not a new exact identity.
No. This calculator is limited to real principal roots. Negative radicands require complex-number notation and methods outside its scope.
About Square Root Calculator
Calculate the principal real square root locally. Perfect-square integers are labeled exact; other displayed roots are explicitly rounded to the selected precision.