Calculate with Square Root
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What is a square root calculator?
The square root of a nonnegative real number x is the unique nonnegative number r whose square equals x. The symbol √x denotes the principal square root; for x > 0, both r and −r solve y²=x, but √x names only the nonnegative root.
Important terms
- Radicand
- The value inside the radical sign; x in √x.
- Principal square root
- The nonnegative number r satisfying r²=x for x≥0.
- Perfect square
- A value whose square root is exact, such as 25 with √25=5.
- Irrational number
- A real number that cannot be written as a ratio of integers; √2 is a common example.
When this method is useful
- Finding a square’s side length from its area.
- Solving equations of the form y²=x while selecting the principal root.
- Computing Euclidean distances and magnitudes.
- Checking whether a value is a perfect square or has an irrational root.
Square Root Calculator
The principal root is the nonnegative side of a square
Formula and variables
- x
- the nonnegative radicand
- r
- the principal, nonnegative square root
- r²
- the check value that should equal x
The square-root operation reverses squaring on nonnegative inputs, choosing the nonnegative root. If no finite decimal r squares exactly to x, the calculator returns a high-precision decimal approximation.
Domain note: The real square root requires x ≥ 0. Inputs are finite decimals or scientific notation with at most 120 digits. A result is exact only when its decimal square reproduces the input; otherwise the display is rounded.
How the formula works
How to use this calculator
- 1
Enter a nonnegative radicand.
- 2
The calculator computes its principal square root, which is never negative.
- 3
A perfect-square decimal returns an exact root; other values show a high-precision approximation.
- 4
Square the displayed root to check how closely it reproduces the radicand.
- 5
When solving y²=x, remember that the equation has roots +√x and −√x when x>0.
Step-by-step examples
Example 1
Problem: Find √25.
Substitution: 5² = 25 and 5 ≥ 0
Result: 5
Five is the principal square root because it is nonnegative and its square is 25.
Example 2
Problem: Find √0.04.
Substitution: 0.2² = 0.04
Result: 0.2
Decimal inputs can have exact square roots too.
Example 3
Problem: Approximate √2.
Substitution: 1.41421356237309504880168872420969807857² ≈ 2
Result: ≈ 1.41421356237309504880168872420969807857
√2 is irrational, so a finite decimal display is rounded.
Worked example, step by step
Rules, edge cases, and related ideas
Nonnegative output
The radical symbol √x denotes the principal root and therefore returns r≥0.
Inverse relationship
For x≥0, (√x)²=x. For any real y, √(y²)=|y|, not always y.
Product property
For nonnegative a and b, √(ab)=√a√b. Domain conditions matter when extending beyond real numbers.
Edge cases and limitations
- √0 is exactly 0.
- Negative real radicands have no real square root and are rejected.
- Perfect-square decimals such as 0.04 return exact finite roots.
- Irrational roots such as √2 require a rounded decimal display.
- The equation y²=x has both signs when x>0, even though √x itself is nonnegative.
How this differs from a related concept
A square root is the principal nonnegative inverse of squaring on [0,∞). Solving y²=x is a separate task: for x>0 its real solutions are y=±√x.
Understand the result
The output is the principal square root, not every solution of y²=x. For a positive radicand, solve the equation by writing y=±√x. Treat a rounded decimal as an approximation in later calculations.
Common mistakes
- Reporting both ±√x as the value of the radical itself.
- Assuming √(y²)=y for negative y; it is |y|.
- Treating an irrational decimal approximation as exact.
- Taking a negative radicand’s square root while expecting a real answer.
Frequently asked questions
What is the principal square root?
It is the unique nonnegative number whose square equals the radicand.
Why is √25 equal to 5 rather than ±5?
The radical symbol denotes only the principal nonnegative root. The equation y²=25 has two solutions, 5 and −5.
Can this calculator find roots of decimals?
Yes. It accepts finite decimal and scientific-notation inputs when the radicand is nonnegative.
Why is √2 not exact as a decimal?
√2 is irrational, so no finite decimal expansion represents it exactly; the calculator displays a rounded approximation.
What is √(x²)?
For every real x, √(x²)=|x| because the principal square root cannot be negative.