What a square root is and why you need it

A square root is the number that, when multiplied by itself, gives you the original number. The square root of 9 is 3, because 3 × 3 = 9. The square root of 16 is 4, because 4 × 4 = 16. You write it with the symbol √, so √9 = 3.

You need square roots when you work backward from an area to find a side length, when you solve certain equations in algebra, or when you calculate distances using the Pythagorean theorem. Most of the time you will use a calculator, but understanding how square roots work and being able to estimate them by hand makes you faster and more confident with the math.

Key Takeaways

  • A square root is the number that multiplies by itself to give the original number; √9 = 3 because 3 × 3 = 9.
  • Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) have whole-number square roots that you can memorize.
  • For non-perfect squares, a calculator gives you the decimal answer in one keystroke.
  • To estimate a square root by hand, find the perfect squares on either side of your number and narrow down from there.
  • The long-division method for square roots works but is slow; use it only if you want to understand the process deeply.

Perfect squares you should memorize

The easiest square roots to work with are perfect squares — numbers whose square roots are whole numbers. If you memorize the first ten, you will recognize them when ready and solve problems faster.

√1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10. Once you know these, you also know that √400 = 20 (because 20 × 20 = 400) and √2500 = 50 (because 50 × 50 = 2500). The pattern holds: if you know the square root of a number, you can find the square root of that number times 100, 10,000, or any power of 100.

Using a calculator to find any square root

For any number that is not a perfect square, a calculator is the practical choice. On a basic scientific calculator, enter the number and press the √ button. On a phone calculator, switch to scientific mode (usually by rotating the phone to landscape), enter the number, and tap the √ symbol.

For example, to find √50, enter 50 and press √. The answer is approximately 7.07. To find √200, enter 200 and press √. The answer is approximately 14.14. The calculator gives you as many decimal places as the screen shows, which is more precision than you need for most real-world problems.

Estimating a square root without a calculator

If you do not have a calculator, you can estimate any square root by finding the perfect squares on either side of it. To estimate √50, you know that √49 = 7 and √64 = 8, so √50 is between 7 and 8. Since 50 is closer to 49 than to 64, √50 is closer to 7 than to 8. A reasonable estimate is 7.1 or 7.2.

To narrow it down further, try squaring a number in the middle. If you guess 7.1, then 7.1 × 7.1 = 50.41, which is slightly too high. If you guess 7.0, then 7.0 × 7.0 = 49, which is slightly too low. So √50 is between 7.0 and 7.1, closer to 7.07. This method — finding the range, then testing numbers within it — works for any square root and requires only multiplication and a sense of which perfect squares are nearby.

The long-division method for square roots

Before calculators existed, people used a pencil-and-paper method similar to long division to find square roots. It is accurate but slow, and you will rarely need it. However, learning it teaches you how square roots actually work and can be useful if you are studying mathematics history or want to understand the process deeply.

The method involves grouping digits in pairs from right to left, finding the largest digit whose square does not exceed the first group, subtracting, bringing down the next pair, and repeating. A full worked example takes several lines of writing. If you want to learn this method, search for "long division method for square roots" and work through a step-by-step example with a specific number. For practical purposes, estimation or a calculator will serve you much better.

Square roots of negative numbers and special cases

In basic mathematics, you cannot find the square root of a negative number, because no real number multiplied by itself gives a negative result. If a problem asks for √−9, the answer does not exist in the real number system. (In advanced mathematics, imaginary numbers handle this, but that is beyond what you need here.)

The square root of 0 is 0, because 0 × 0 = 0. The square root of 1 is 1, because 1 × 1 = 1. For very large numbers, the same rules explore: √10,000 = 100, √1,000,000 = 1,000. If you see a square root symbol with a small number in front of it — like ³√8 — that is a cube root (the number that multiplies by itself three times), not a square root, and it works differently.

Common mistakes and how to avoid them

A common mistake is thinking that √9 = 3 and 3 is the only answer. In fact, (−3) × (−3) = 9 as well, so technically √9 has two answers: 3 and −3. However, by convention, the √ symbol means the positive square root only, so √9 = 3. If a problem asks for "all square roots of 9," the answer is ±3 (plus or minus 3), but if it asks for √9, the answer is 3.

Another mistake is confusing √(a + b) with √a + √b. They are not the same. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Always compute what is inside the square root symbol first, then take the square root of that result.

Frequently Asked Questions

What is the square root of 2?

The square root of 2 is approximately 1.414. It is not a whole number or a straightforward fraction; it is an irrational number, meaning its decimal digits go on forever without repeating. You will see it written as √2 ≈ 1.414 or 1.41, depending on how much precision you need.

How do I find the square root of a decimal number?

Use a calculator the same way you would for a whole number. Enter 2.25 and press √, and you get 1.5 (because 1.5 × 1.5 = 2.25). For estimation by hand, treat it the same way: find perfect squares on either side and narrow down by testing.

Is there a difference between √9 and 9^(1/2)?

No. The notation 9^(1/2) means "9 to the power of one-half," which is another way of writing the square root. Both √9 and 9^(1/2) equal 3. The exponent notation is useful in algebra and on calculators that do not have a √ button.

Can I simplify a square root like √50?

Yes. √50 = √(25 × 2) = √25 × √2 = 5√2. This is useful in algebra when you want to leave the answer in exact form rather than as a decimal. You can simplify any square root by factoring out perfect squares from inside the radical.

Why do I need to know square roots if I have a calculator?

Understanding square roots helps you check whether a calculator answer makes sense, estimate quickly in your head, and solve algebra problems where the answer is left in radical form. It also builds your number sense and confidence with mathematics.