Square Root Calculator

Enter a number. You get its square root, a simplified radical, and a perfect-square check.

Smart Tips

• The radical sign means the principal, non-negative square root. √16 is 4, not −4.

• A whole number is simplified by pulling out the largest perfect-square factor. √50 = 5√2.

• A perfect square has a square-free part of 1, so the radical disappears. √16 = 4.

• Decimals get a numeric root. The integer-times-square-free form is reserved for whole numbers.

The Formula

√(k² · n) = k√n

√m = Principal root
The non-negative number whose square is m

k = Outside factor
The root of the largest square factor

n = Square-free part
What is left under the radical

m = Radicand
The number inside the radical

A square root is simplified when the radicand has no perfect-square factor other than 1. The page also says whether the number is a perfect square. For a decimal, it checks whether the reduced fraction is a ratio of perfect squares.

Key Insights

Not the keypad

The scientific calculator can evaluate a square root. This page also factors the radicand and says whether it is a perfect square.

Product property

√(ab) = √a · √b for non-negative a and b. That is how √50 becomes √25 · √2, then 5√2.

Zero and negatives

√0 = 0, and 0 is a perfect square. A negative number has no real square root, so the page rejects it.

Perfect squares

0, 1, 4, 9, 16, 25, and so on simplify to integers. 2.25 is also a perfect square, because it is (3/2)², but it is not written as an integer radical.

Frequently Asked Questions

The principal square root, which is the non-negative number whose square is the value you entered. √16 is 4. The negative square root is a separate number, written −√16.

An integer times a square-free whole number. The page finds the largest perfect-square factor, takes its root outside the radical, and leaves the square-free part inside. √50 = 5√2. √16 = 4, because 16 is a perfect square.

The integer-times-square-free form is for whole numbers. A decimal still gets the principal square root and a perfect-square check. 2.25 is a perfect square because 2.25 = 9/4 and both 9 and 4 are perfect squares.

No. A negative number has no real square root. The page asks for zero or a positive number. Zero is allowed, and its square root is zero.

For a whole number, the square-free part is 1. For a decimal, the page writes the number as a reduced fraction and checks whether the numerator and the denominator are both perfect squares.

The principal square root and perfect squares follow OpenStax Elementary Algebra 2e, section 9.1. Simplifying with the product property, so that a radical is an integer times a square-free factor, follows section 9.2 of the same book. No outside data feed is used.

Responsive Ad Space

Square root

Enter values to calculate

Simplified radical

Perfect square

Note:The radical sign means the principal (non-negative) square root. A simplified radical is an integer times a square-free whole number, and it is shown for whole-number radicands. Negative numbers are rejected.