Square roots and square numbers show up everywhere in school math, from geometry homework to algebra exams, so calculating square roots is one of those skills that pays off across many subjects. Whether you are just getting started or want to learn how to calculate a square root by hand without a calculator, this guide walks you through everything you need to know.
You will learn five methods for finding the square root of a number, from the quick calculator approach to the step-by-step long division method that works for any number, perfect square or not. If square roots feel like a confusing concept right now, that is completely normal; by the end of this guide, they will not be.
What Is a Square Root?
The square root of a number is the value you multiply by itself to get the original number. In other words, it is the value that, when multiplied by itself, produces the original.
- If 3 times 3 equals 9, then the square root of 9 is 3.
Finding the square root of a number is the inverse operation of squaring a number. If squaring means multiplying a number by itself, taking the square root reverses that process.
The Square Root Symbol (√)
The symbol used to represent a square root is called the radical symbol (also known as the radical sign), written as √. It likely evolved from the letter “r” for the Latin word radix, meaning root. The number placed under the radical sign is called the radicand. So in the expression √36, the radical symbol is √ , and the radicand is 36. The result, 6, is the square root.
You will also sometimes see square roots written using exponent notation: √n is the same as n^(1/2). Both forms mean exactly the same thing, though the radical symbol is far more common in school settings.
Perfect Squares and Non-Perfect Squares
Perfect Squares
A perfect square is a number whose square root is a whole number. These are numbers produced when an integer is squared:
- 1 squared = 1
- 2 squared = 4
- 3 squared = 9
and so on.
The numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all perfect squares. The square root of a perfect square is always a clean integer (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
Non-Perfect Squares
A non-perfect square (also called an imperfect square) is any positive number that does not have a whole number as its square root. The square root of a non-perfect square, like 2 or 7, is an irrational number: the decimal never terminates and never repeats. The method you choose often depends on whether you are working with a perfect square or a non-perfect square, so it is useful to recognize the difference before you start.
5 Methods to Calculate a Square Root
There is no single best method for computing square roots. The right approach depends on what tools you have, whether the number is a perfect square, and how much precision you need.
Here are five methods explained step by step, ordered from simplest to most involved.
Method 1: Using a Calculator (Easiest)
The easiest way to find a square root is with a scientific calculator. Most calculators have a dedicated square root button, usually labelled with the radical symbol √ or written as “sqrt.” To use it, simply enter the number you want to find the square root of, then press the square root button. The calculator displays the result instantly.
For example, to find √144:
- Enter 144 on the calculator.
- Press the √ button.
- The display shows 12, which is the exact answer.
For non-perfect squares like √50, the calculator will return a decimal approximation (approximately 7.071). This is accurate to several decimal places and is usually precise enough for homework and exams.
Method 2: The Repeated Subtraction Method
The repeated subtraction method is one of the most intuitive ways to find the square root of a perfect square. The idea is simple: you subtract consecutive odd numbers (1, 3, 5, 7…) from your number, one at a time, until you reach zero. The number of subtractions you performed is the square root.
Here is a worked example using √16:
- Step 1: 16 – 1 = 15
- Step 2: 15 – 3 = 12
- Step 3: 12 – 5 = 7
- Step 4: 7 – 7 = 0
You subtracted four times, so √16 = 4.
This method works only for perfect squares and becomes tedious for larger numbers. But it is a great way to understand the concept of square roots because it makes the relationship between odd numbers and perfect squares concrete and visual.
If your child needs extra support working through these fundamentals, private tutoring can help them build a solid foundation at their own pace.
Method 3: The Prime Factorization Method
The prime factorization method uses a number’s prime factors to calculate its square root. Every whole number can be expressed as a product of prime numbers, and this structure makes the square roots of perfect squares predictable.
Here are the steps to find the square root using prime factorization:
- Step 1: Factorize the number into its prime factors.
- Step 2: Pair up identical prime factors.
- Step 3: Take one factor from each pair.
- Step 4: Multiply the selected factors together to get the square root.
Let us find √144 using this method:
- 144 = 2 × 2 × 2 × 2 × 3 × 3
- Pairs: (2 × 2) and (2 × 2) and (3 × 3)
- Take one from each pair: 2, 2, and 3
- Multiply: 2 × 2 × 3 = 12
So √144 = 12.
This method is precise and elegant for perfect squares, but it does not work directly for imperfect squares. For those, the estimation or long division methods below are better choices.
H3: Method 4: The Estimation Method
The estimation method (also called the guess-and-check method) is a practical approach for finding an approximate square root of a non-perfect square, particularly when a calculator is not available. It works by narrowing down the answer using the nearest perfect squares you already know.
To estimate √50:
- Find the nearest perfect square below: √49 = 7. Find the nearest perfect square above: √64 = 8. So √50 falls between 7 and 8.
- Since 50 is much closer to 49 than to 64, the root is closer to 7.
- Try 7.1: 7.1 × 7.1 = 50.41 (a little high)
- Try 7.07: 7.07 × 7.07 = 49.98 (very close)
So √50 is approximately 7.07, which you can confirm with a calculator (the exact decimal is 7.0710…).
This guess-and-check approach builds genuine number intuition and is especially useful during exams when you need a quick, reasonable approximation rather than a digit-perfect result.
Understanding estimation is also a gateway to deeper mathematical thinking that helps students across algebra, geometry, and beyond.
Method 5: The Long Division Method (Most Accurate)
The long division method is the most accurate way to find a square root by hand, and it works for any positive number. If you need to calculate a square root without a calculator, this is the method to learn. It produces exact results for perfect squares and as many decimal places as you need for imperfect ones.
Here is the step-by-step process for finding √529:
- Step 1: Group digits in pairs from right to left using bar notation: 5 | 29. Each pair of digit pairs will produce one digit of the answer.
- Step 2: Find the largest integer whose square is less than or equal to the leftmost group (5). That number is 2, because 2 × 2 = 4.
- Step 3: Write 2 above the bar. Subtract 4 from 5; the remainder is 1.
- Step 4: Bring down the next pair (29), making the new dividend 129.
- Step 5: Double the current quotient (2 × 2 = 4). Write 4_ as the start of the new divisor.
- Step 6: Find the digit N such that (40 + N) × N is less than or equal to 129. Testing N = 3: 43 × 3 = 129. It fits exactly.
- Step 7: Write 3 above the bar. Subtract 129 from 129; remainder is 0.
Result: √529 = 23.
Once you work through a few examples, the pattern becomes clear. For non-perfect squares, add pairs of zeros after the decimal point and continue the process to produce as many accurate decimal places as needed.
Square Roots Reference Table
Memorizing the square roots of the first several perfect squares saves time on tests and builds numerical fluency. Here are the square roots for whole numbers 1 through 15:
| Number | Square Root |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
| 121 | 11 |
| 144 | 12 |
| 169 | 13 |
| 196 | 14 |
| 225 | 15 |
Numbers that are not perfect squares (like 2, 3, 5, and 7) produce irrational numbers when square-rooted. Their decimal values never terminate and never repeat, which is why a calculator or the long division method is needed for accurate results.
How to Simplify a Square Root
Sometimes you are not asked to calculate a square root numerically but rather to simplify a radical expression. For example, √50 in simplified form is 5√2, not just the decimal 7.071.
To simplify a square root:
- Find the factors of the number under the radical symbol.
- Identify any perfect square factors.
- Take the square root of the perfect square factor and move it outside the radical.
- Leave any remaining factor inside the radical.
For √50:
- Factors of 50 include 25 × 2
- 25 is a perfect square: √25 = 5
- So √50 = √(25 × 2) = 5√2
For √72:
- 72 = 36 × 2
- √36 = 6
- So √72 = 6√2
The goal of simplification is to produce the simplest possible form of a radical expression, which makes further calculations cleaner. Using prime factorization to find factors is often the most reliable approach, especially when the number is large.
Positive and Negative Square Roots
Most of the time, when you are asked to calculate the square root of a number, you will give a positive answer. However, every positive number technically has two square roots: one positive and one negative.
Here is why: If you multiply 5 by itself, you get 25, and if you multiply -5 by itself, you also get 25 (because a negative number times a negative number gives a positive result). So both 5 and -5 are technically square roots of 25.
In practice, when you see the radical symbol (√), it refers to the principal square root, which is always the positive value. So √25 = 5, not -5. The negative square root would be written as -√25 = -5. It is also worth knowing that the square root of a negative number is not a real number; it produces an imaginary number (written using the symbol i).
For most school problems and everyday calculations, you will work with positive results only.
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Real-World Applications of Square Roots
Square roots are not just an abstract school topic. They appear in real calculations across several fields, and understanding them gives students a concrete reason to work through the harder methods.
Square Roots in Finance
One practical use of the square root formula is calculating investment return rates. The compound annual growth rate (CAGR) formula for a two-year period uses a square root to find the annualized return. If an investment grows from $1,000 to $1,210 over two years, the square root of 1.21 (which is 1.1) tells you the annual growth rate was 10%. Notice that 1.21 is one of the square numbers, and its root is the clean value 1.1.
Square Roots in Geometry and Construction
The Pythagorean theorem, one of the most famous equations in mathematics, relies directly on square roots. The theorem states that in a right-angle triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². To find the length of the hypotenuse, you take the square root of (a² + b²). This calculation is used constantly in construction and building work to check that angles are true and measurements are correct.
If a room is 3 metres wide and 4 metres long, the diagonal distance across it is √(9 + 16) = √25 = 5 metres. Contractors use this kind of calculation every day.
Square Roots in Statistics
In statistics, standard deviation is defined as the square root of variance. Variance measures how spread out a set of numbers is, and standard deviation expresses that spread in the same units as the original data, which makes it far more interpretable. Understanding this relationship is essential for high school statistics courses and becomes even more important in university-level data analysis.
For students taking science tutoring courses that include data interpretation, a strong grasp of square roots makes the statistical sections significantly more manageable.
Still Struggling With Square Roots? Tutorax Can Help
Square roots are one of those topics that seem straightforward until the methods get more involved. The long division method, in particular, trips up many students, and it is rarely taught with enough worked examples to feel comfortable. If your child is struggling with math or simply needs more practice time than a classroom setting allows, one-on-one support can make a real difference.
At Tutorax, we have helped more than 20,000 students across Canada improve their academic performance, and 95% of them see measurable grade improvements. Our tutors work one-on-one with each student, helping them improve math skills and build confidence in areas like square roots, algebra, and beyond.
Here is what working with a Tutorax tutor looks like:
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Whether your child needs help catching up or wants to get ahead before exams, find a math tutor in your area today. We offer both online tutoring and in-home tutoring across all Canadian provinces, including dedicated tutoring in Toronto and math tutors in Montreal.
Every student can master square roots with the right support. The methods in this guide are a great starting point, and a qualified tutor can walk through each one in a way that actually sticks.
Not sure what tutoring will cost? See our breakdown of how much a math tutor costs in Canada.