Square Root Calculator (√x)

What is a Square Root?

A square root is a number that gives the original value when multiplied by itself. For example, because $12 \times 12 = 144$, the principal square root of 144 is $\sqrt{144} = 12$. The square root symbol is called a radical sign, and square roots can be written in radical or exponent form.

Square Root Formula

If $y$ is the principal square root of a non-negative number $x$, then:

A square root can also be written as a fractional exponent: $\sqrt{x} = x^{1/2}$.

How to Calculate a Square Root?

1. Step 1: Determine whether the number is a perfect square.
2. Step 2: If it is not a perfect square, factor the number and look for pairs of equal factors.
3. Step 3: Move one factor from each pair outside the radical.
4. Step 4: Keep any unpaired factors inside the radical and simplify the result.
5. Step 5: For a decimal answer, evaluate the simplified radical to the required precision.

Square Root Examples

Negative Numbers and Variables

Square roots behave differently depending on whether the input is positive, zero, or negative. A negative number does not have a real square root, but its square root can be expressed using the imaginary unit $i$, where $i = \sqrt{-1}$.

Where Are Square Roots Used?

Common Square Root Mistakes

• Splitting Addition: In general, $\sqrt{a+b} \ne \sqrt{a}+\sqrt{b}$. For example, $\sqrt{25}=5$, while $\sqrt{9}+\sqrt{16}=7$.
• Confusing Root and Equation Solutions: $\sqrt{16}=4$ means the principal square root is 4. However, the equation $x^2=16$ has two solutions: $x=\pm4$.
• Stopping Too Early: A radical such as $\sqrt{72}$ can still be simplified to $6\sqrt{2}$ because 72 contains the perfect-square factor 36.