Chapters
What Is a Square Root?
In simple terms, finding the square root of a number is the exact opposite (the "inverse") of squaring a number.
If you square the number 5, you multiply it by itself: 5 × 5 = 25
Therefore, the square root of 25 is 5. It is the "root" factor that was used to grow the larger number.
Formal Definition: A square root of a number, a, is a number, b, which, when multiplied by itself (elevated to the square), is equal to the original number:
Square Root Notation (√)
In square root maths, we use a special symbol called a radical to show we are looking for a root.
The Symbol: √
When you see the symbol √, it is telling you to find the number that multiplies by itself to get the value inside.
- Radicand: This is the number sitting inside the symbol. In √36, 36 is the radicand.
- Index: This is the small number that usually sits in the "V" of the symbol to tell you which root to find. For a square root, the index is 2. However, because square roots are so common, the index is almost always omitted. If you see a radical with no number, assume it is a 2.
Perfect Squares
A perfect square is a number that has an exact square root (a whole number). These are the most important numbers to learn when you are starting with square root for beginners.
Common Perfect Squares List
Here are the first twelve perfect squares that every student should know:
| Number | Multiplication | Perfect Square |
|---|---|---|
| 1 | 1 × 1 | 1 |
| 2 | 2 × 2 | 4 |
| 3 | 3 × 3 | 9 |
| 4 | 4 × 4 | 16 |
| 5 | 5 × 5 | 25 |
| 6 | 6 × 6 | 36 |
| 7 | 7 × 7 | 49 |
| 8 | 8 × 8 | 64 |
| 9 | 9 × 9 | 81 |
| 10 | 10 × 10 | 100 |
| 11 | 11 × 11 | 121 |
| 12 | 12 × 12 | 144 |
Square Roots of Non-Perfect Squares
If a number is not a perfect square (like 2, 3, or 10), its square root is not exact.
If you try to find √10, the answer will be a decimal that never ends and never repeats (an irrational number).
For these numbers, we usually provide a simple decimal approximation or leave them in their "surd" form (keeping the root symbol).
Positive and Negative Square Roots
Technically, most numbers have two square roots: a positive one and a negative one.
Take 25 as an example:
- 5 × 5 = 25
- (-5) × (-5) = 25
However, in most school maths, when you see the symbol 
, we are looking for the principal square root, which is the positive answer (5). If a question wants both, it will usually use the symbol 
.
Worked Examples
Example 1: Basic Calculation
Find 
.
- Step 1: Think of your times tables. What number multiplied by itself equals 81?
- Step 2: We know 9 × 9 = 81.
- Answer: 9
Example 2: Square Root of a Fraction
Find 
.
- Step 1: Find the square root of the top number (numerator):
. - Step 2: Find the square root of the bottom number (denominator):
. - Answer:
Example 3: Area of a Square
A square garden has an area of 100 square metres. How long is one of its sides?
- Step 1: The area of a square is side x side.
- Step 2: To find the side length, find the square root of the area.
- Step 3:
. - Answer: 10 metres.
Common Mistakes to Avoid
- Confusing Squares with Square Roots: Many students see
and write 256. Remember: Square root makes the number smaller (unless the number is between 0 and 1). - Splitting Additions:
is not the same as 
. - Correct:
. - Incorrect: 3 + 4 = 7.
- Correct:
- Negative Radicands: You cannot find the square root of a negative number using real numbers, because any number multiplied by itself (positive or negative) results in a positive.
Practice Questions & Solutions
What is the key definition of a square root?
A number that when multiplied by itself gives the original number.
Identify the perfect squares in this list: 4, 12, 20, 49, 75, 121.
4, 49, 121
Calculate 
.
12
If 
, what is the value of 
?
8 (or ±8)
Estimate whether 
is closer to 4 or 5.
Closer to 4 (since 4 squared is 16 and 5 squared is 25)
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I’m afraid this article (on cardinal numbers) has an unfortunate error. Zero must be considered a cardinal number. The definition of a cardinal number given by the Penguin Dictionary of Mathematics is “A number that indicates the number of elements in a set.” It also defines a null set or empty set containing no elements. The cardinality of this set is 0.