Square Root Calculator
Calculate square roots and related values
Instant calculations with precise formulas. No complexity, just the results you need.
Formula
√xHow to Use Square Root Calculator
- Provide your input values.
- Understand the calculation method shown.
- Get immediate and accurate results.
- Use the output for planning or analysis.
Examples
A square root asks which number, multiplied by itself, gives the value under the radical, and the answer is only a whole number for a small, specific set of inputs called perfect squares; everything else has a square root that goes on forever without repeating.
The formula
sqrt(x) = r such that r x r = x, for x >= 0- x
- the number you are taking the root of
- r
- the square root, the result
Worked example
sqrt(49) = 7 exactly, because 49 is a perfect square, 7 x 7. sqrt(50) has no exact answer; it works out to approximately 7.0710678, a decimal that never terminates or repeats, because 50 sits between the perfect squares 49 and 64 and has no whole number root. sqrt(-49) has no real answer at all, since squaring any real number, positive or negative, always produces a positive result, so no real number multiplied by itself gives negative 49.
What trips people up
- Every positive number actually has two square roots, positive and negative, since (-7) x (-7) also equals 49; the square root symbol by convention returns only the positive one.
- A negative number under the root is not an error to be rounded away, it genuinely has no real square root, only an imaginary one written with i.
- sqrt(a) + sqrt(b) is not the same as sqrt(a + b); sqrt(9) + sqrt(16) is 7, while sqrt(25) is only 5.
- Only perfect squares, 1, 4, 9, 16, 25, and so on, produce whole number roots; every other positive integer has an irrational one.
Frequently asked questions
- Why does the square root symbol only give a positive answer?
- By mathematical convention, the radical sign is defined to return the principal, non-negative root, even though a negative root also technically satisfies r x r = x.
- What is an imaginary number and when does it appear here?
- It appears whenever you take the square root of a negative number; sqrt(-49) is written 7i, where i is defined as sqrt(-1), since no real number squared gives a negative result.
- How can I estimate a square root without a calculator?
- Find the two perfect squares it sits between; sqrt(50) is between sqrt(49) = 7 and sqrt(64) = 8, and since 50 is close to 49, the root is close to 7, around 7.07.
- Is a cube root calculated the same way as a square root?
- Related but not identical; a cube root asks which number cubed gives x, and unlike a square root, a cube root of a negative number is a real, negative number, since a negative number cubed stays negative.
- What is the square root of 0?
- 0, since 0 x 0 = 0, and it is the one nonnegative number whose positive and negative square roots are the same value.