Find principal real or complex square roots, test exact rationality and inspect radical simplification, square grids and approximations.
01 / PURPOSE-SPECIFIC INPUTS
02 / RESULT & INTERACTIVE INSPECTION
Perfect-square status means a nonnegative integer square. Rational decimals such as 0.25 can have exact rational square roots without being integer perfect squares. Large residual radicals are not falsely labeled fully simplified.
Trial 8.5 squared = 72.25; target magnitude 72. Exact form: 6 × √2.
Enter your values locally; no inputs are uploaded.
Check validation feedback before reviewing the browser-generated result.
Review the explanation, copy supported outputs or reset the workspace.
The principal real square root is the unique nonnegative root. Solving y²=x has two signs for nonzero x, but that equation-solving convention is distinct from the primary √x result.
Find principal real or complex square roots, test exact rationality and inspect radical simplification, square grids and approximations. Inspect actual values and worked assumptions through accessible interactive controls.
Square Root Calculator calculations run locally in browser memory. Expressions, measurements, schedules and datasets are not sent to an external calculation-processing API or saved as a personal profile. Ordinary website requests may still be logged. Clipboard access occurs only when you choose Copy Result.
The principal real square root is the unique nonnegative root. Solving y²=x has two signs for nonzero x, but that equation-solving convention is distinct from the primary √x result.
The square factor 36 is extracted: √72 = √(36×2) = 6√2. The decimal is separately approximate, about 8.485281; the radical remains the exact value.
The reduced exact rational numerator and denominator are tested with integer square roots. Both must be perfect squares for a rational root magnitude; otherwise the magnitude is irrational, including √2.
The principal complex result is i times the positive magnitude root. For example √−9 = 3i. Its magnitude diagram is labeled as an imaginary-component magnitude rather than a real-axis root.
No. Exact perfect-square tests run first, while complete trial square-factor extraction is bounded to numerator/denominator values at most 10¹⁰. Larger residual radicals remain exact but are explicitly not claimed fully simplified.
Its exact root is 1/2. The separate perfect-square label means a nonnegative integer square, while rational decimals/fractions can have rational roots without being integer perfect squares.