Exponents describe repeated multiplication, roots undo a power relationship, and logarithms name the exponent needed to reach a value.
Use it to move transparently between a base, exponent, root, logarithm, and equivalent notation.
The relationship
Write the rule before calculating
√x is the nonnegative number whose square is x.
See the structure
What the calculation is doing
2³ = 8√x is the nonnegative number whose square is x.
Worked interpretation
Read the result in context
√81 = 9.
Interpret with care
Important boundary
Even roots require a non-negative real radicand here, while logarithms require a positive argument and valid base.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Understand the subject
What is a square root?
The principal square root of a nonnegative number is the nonnegative value whose square returns that number.
Use square roots to reverse squaring in geometry, distance, variance, scaling, and algebra.
The relationship
The defining equation
√x = r exactly when r²=x and r≥0
x is the radicand and r is the principal root.
See the structure
A square’s area reveals its side
area 8199√81 is the side length: 9
Worked example
Find √81
9²=81, so √81=9
Interpret with care
What the result does—and does not—mean
Although both 9 and −9 solve r²=81, the symbol √81 means the principal nonnegative root 9.
Quick guide
How to use this calculator
Choose a calculation mode when the page offers more than one relationship.
Enter the labelled known values.
Read the primary result and use the check or equivalent form to verify its meaning.
Calculation method
Apply the exponent, root, or logarithm relationship
√x is the nonnegative number whose square is x.
The calculator checks domains, zero denominators, unsupported powers, branches, and finite-range limits before reporting a result.
Worked example
Worked example
√81 = 9.
√x is the nonnegative number whose square is x.
Supported inputs
Precision and limits
Real-number scope
This calculator returns real results and reports inputs outside its stated real domain.
Numerical range
Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. Exact integer radical simplification uses its narrower disclosed bound.
Precision
Numerical functions use double-precision arithmetic and normally display 12 significant digits. Symbolic labels are exact only where explicitly stated.
