What is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number.
In other words, if you know that 6 × 6 = 36, then 6 is the square root of 36.
Every positive number has two square roots—one positive and one negative—because both can be squared to get the same result. For example, both 5 and -5 are square roots of 25.
But when people say “the square root,” they usually only mean the positive value. Mathematicians refer to the positive value as the principal square root, which we’ll use throughout this post.
We use the square root symbol (√) to represent them. For example:
√25 = 5
The number under the square root symbol is called the radicand. In √25, the radicand is 25.
You might also see them written with exponents. The square root of a number is the same as raising it to the power of 1/2. So:
√x = x¹ᐟ²
Example:
√121 = 121¹ᐟ² = 11
? Quick Tip: Square Roots and PEMDAS
Because a square root is the same as an exponent (power of ½), it falls under the “E” in PEMDAS order of mathematical operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Learn more about PEMDAS: Mathematical Order of Operations.
In this post, you will learn about square roots, why they matter, and how to find and simplify them. We’ll even take a quick look at the square roots of negative numbers, which sound impossible but have practical uses!
Perfect vs. Non-Perfect Squares
Perfect squares are numbers like 1, 4, 9, 16, 25, and so on. These are numbers that come from squaring natural numbers (12, 22, 32, 42, 52. . .).
A perfect square is a number that comes from multiplying a whole number by itself. For example:
Perfect squares always have whole numbers as their square roots, like √25 = 5.
Finding the square root of a perfect square is easy. For example:
- √49 = 7 (because 7 × 7 = 49)
- √100 = 10 (because 10 × 10 = 100)
On the other hand, a non-perfect square is any number that isn’t a perfect square—like 2, 3, 5, 10, or 50. The square roots of these numbers are irrational. An irrational number is one that has a decimal form that goes on forever without repeating, such as pi (π). For example:
Because this decimal never ends or repeats, we round it for estimation.
Most numbers are not perfect squares. That’s because there is only one perfect square for each whole number. So, perfect squares get farther apart as numbers get larger. There are many non-perfect squares between every pair of perfect squares, so non-perfect squares are much more common overall.
Understanding the difference helps you decide whether a square root will be a clean whole number or an endless decimal.
Learn more about Irrational Numbers: Definition & Examples.
Why Do Square Roots Matter?
You might be wondering how square roots are useful outside of math class. The answer is—they show up all over the real world.
In construction and design, they are used to calculate areas and dimensions. For example, if an architect knows the area of a square room is 100 square feet, they’ll use the square root to figure out each side: √100 = 10 feet.
In science and engineering, they appear in energy, force, and motion formulas. If you’re measuring how far something moves or how fast it accelerates, square roots help describe the relationship between those values.
In computer graphics and video games, they help calculate distances between objects. When a character moves across a 3D space, the game engine often uses the distance formula—which includes a square root—to figure out how far the character needs to go.
The square root of the product of two numbers yields the geometric mean of those two numbers. Learn more about the Geometric Mean: Definition, Formula & Finding.
In statistics, some of the most important formulas include square roots. Here are a few examples:
- The standard deviation tells us how spread out data are and it is just the square root of another statistic, the variance.
- The standard error formula indicates the precision of a sample estimate and helps evaluate statistical significance and construct confidence intervals. The square root in the denominator defines how sample estimates become more precise as the sample size increases.
Analysts use these statistical tools in everything from scientific research to sports analysis to business forecasting. That means when you’re studying square roots, you’re building the foundation for understanding data and making accurate decisions in the world.
How to Find the Square Root of a Number
There are several ways to find the square root of a number. Some work best for perfect squares, while others help with numbers that don’t have clean roots. Here are three of the most common methods.
Estimation Method
Use this method when the number isn’t a perfect square.
Example: Find √30
Step 1: Identify nearby perfect squares.
√25 = 5 and √36 = 6, so √30 is between 5 and 6.
Step 2: Try a number between 5 and 6.
Try 5.5 → 5.5² = 30.25 (a little too high)
Try 5.4 → 5.4² = 29.16 (a little low)
Step 3: Narrow it down.
Try 5.48 → 5.48² = 30.03
So √30 ≈ 5.48
This method is quick and helps you get a decimal approximation.
Prime Factorization Method
This method works well for perfect squares.
Example: Find √100
- Break 100 into prime factors: 100 = 2 × 2 × 5 × 5
- Group the pairs: (2 × 2), (5 × 5)
- Take one number from each pair: 2 × 5 = 10
So √100 = 10
Learn more about Prime Numbers and Factors.
Repeated Subtraction Method (Perfect Squares Only)
This simple method works only for perfect squares.
Example: Find √36
Subtract consecutive odd numbers until you reach zero:
36 – 1 = 35
35 – 3 = 32
32 – 5 = 27
27 – 7 = 20
20 – 9 = 11
11 – 11 = 0
You subtracted six times, so √36 = 6
Bonus: Long Division Method
If you’re curious or want a more exact answer without a calculator, you can use the long division method. It works for any number and gives as many decimal places as you need.
See the bonus section at the end of this post to see how it works with √200.
How to Simplify Square Roots
Simplifying a square root means rewriting it in a cleaner form. It doesn’t change the value—it just makes it easier to work with when solving equations or combining square root terms. It’s like reducing a fraction: the simplified version is cleaner and often more useful. It also keeps answers exact instead of switching to decimals.
To simplify square roots, break the number into prime factors and look for pairs of the same prime factors. When there is a pair of prime factors, take one outside and delete the other.
Example: Simplify √50
- 50 = 2 × 5 × 5
- Pair the 5s: (5 × 5)
- Take one 5 out and delete the other: √50 = 5√2
Example: Simplify √72
72 = 2 × 2 × 2 × 3 × 3
Take one number from each pair: 2 and 3
√72 = 6√2
You can also simplify square roots in fractions:
√(49/4) = √49 / √4 = 7 / 2
Simplifying square roots also helps when comparing values, doing mental math, or estimating values.
? Quick Example: Simplifying Helps Estimation
You can simplify:
If you know that:
Then you can estimate:
So √200 ≈ 14.14 — the same result you’ll get from the long division method that I show later.
This example shows how simplifying a square root can sometimes make estimation faster and easier!
Square Root Table for Numbers 1 to 10
A square root table can be helpful when learning about them. It’s a quick way to see the square roots of small numbers. You can also use it with simplified square roots to estimate other values quickly.
| Number | Square Root |
| 1 | 1.000 |
| 2 | 1.414 |
| 3 | 1.732 |
| 4 | 2.000 |
| 5 | 2.236 |
| 6 | 2.449 |
| 7 | 2.646 |
| 8 | 2.828 |
| 9 | 3.000 |
| 10 | 3.162 |
Use this table when estimating or checking your answers.
Square Root of Negative Numbers
At first, it might sound strange—or even impossible—to take the square root of a negative number. After all, no real number multiplied by itself gives a negative result.
That’s where imaginary numbers come in. The square root of -1 is defined as i, which stands for “imaginary unit.”
So:
√(-49) = √49 × √(-1) = 7i
Even though they’re called “imaginary,” these numbers have real-world uses. Engineers use them to model electrical circuits. Physicists use them in quantum mechanics. Computer scientists use them when working with signals and algorithms.
Imaginary numbers might seem odd at first, but they help us describe things in the real world that regular numbers can’t. They’re a powerful part of advanced math.
Bonus: Long Division Method of Finding Square Roots
This method is for students who want to dig a little deeper. While it’s not commonly taught today, it’s a great way to see how square root values are built by hand—digit by digit. It strengthens your understanding of place value, reinforces estimation skills, and shows you where those long decimals come from. It’s also useful when calculators aren’t allowed, and it gives you a powerful sense of how math works behind the scenes.
The long division method helps you find the square root of any number, even if it’s not a perfect square. It works by breaking the number into pairs of digits and finding one digit at a time.
Let’s find the square root of 200 step by step.
Step 1: Place a bar over every pair of digits from right to left
For 200, we group the digits as: 2 | 00
Each bar tells us how many digits we’ll work with at a time.
Step 2: Find the largest number whose square is less than or equal to the first group (2).
1 × 1 = 1 fits.
Write 1 in the quotient (this becomes part of your answer).
Subtract 1² = 1 from 2. The remainder is 1.
Step 3: Bring down the next pair of digits.
Now bring down the next pair (00), making the new dividend 100.
Double the current quotient (1) to get 2. This value becomes the start of a new trial divisor.
To complete the divisor, we look for a digit x that we can add to 2 (making 20 + x), so that when we multiply by x, the result is less than or equal to 100.
This idea is the key to the long-division method:
You build the new divisor by placing a digit x next to the doubled quotient, then multiply the full value by x to test it.
Try x = 4:
That works.
Write 4 in the quotient. Now the quotient is 1.4
Subtract: 100 – 96 = 4
Step 4: Bring down the next pair of zeros.
Now the new dividend is 400.
Double the current quotient (14) to get 28, which becomes the base of the next trial divisor.
Now find a digit x such that:
Try x = 1:
That fits.
Write 1 in the quotient. Now the quotient is 1.41
Subtract: 400 – 281 = 119
Step 5: Continue the process by bringing down another pair of zeros.
The new dividend is now 11,900.
Double the current quotient (141) to get 282, then try digits x that fit:
Try x = 4:
That works. Add 4 to the quotient: now it’s 1.414
You can repeat this process to find as many decimal places as you want.
Step 6: Read the result.
So far, we have:
You can continue the long division to get more precise digits if needed.
Comments and Questions