Squares and Square Roots

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Adding and Subtracting Multi-Digit Whole Numbers Worksheets

The square of a number?

When we multiply an integer by itself we call the product the square of the number.
For example: 4 x 4 = 16
The square of 4 is 16
The pictures above show why we call these products squares.

16 units arranged 4 by 4

Squared Numbers

The examples below show how squares can be found.

Number Multiplied by
itself
Square Number Multiplied by
itself
Square
1 1 x 1 1 9 9x9 81
2 2 x 2 4 10 10x10 100
3 3 x 3 9 11 11x11 121
4 4 x 4 16 12 12x12 144
5 5 x 5 25 13 13x13 169
6 6 x 6 36 14 14x14 196
7 7 x 7 49 15 15x15 225
8 8 x 8 64
 12 x 12 multiplication grid with squares highlighted

Squared Words!

You'll hear different words used when people talk about squares. e.g.

  • Square of a number: 25 is the square of 5
  • Squaring a number: multiplying the number by itself
  • A squared number: 100 is a square number
  • squared: 3 squared is 9
  • What's the square of 9? The square of 9 is 81
  • Perfect square: Perfect square is another term for square number

Make sure your children know the short way of finding a square on a calculator. e.g. to find 6 x 6, enter 6, x, =

Writing squares

Squares are powers of two.

We write squares using the same notation that we use with other powers.
e.g. for 3 squared (or 3 x 3)
we write 32

We could talk about 3 to the power of 2, or the second power of three but we don't usually do so; we say 3 squared. We write 32

Square Roots

You can think of finding square roots as the opposite of finding squares. You find the square root of a number (let's call it number A) by finding the number that, when multiplied by itself produces number A. The examples below show this:

Number Square Root Number Square Root
1 x 1 = 1 1 9x9 81 9
2 x 2 = 4 2 10x10 100 10
3 x 3 = 9 3 11x11 121 11
4 x 4 = 16 4 12x12 144 12
5 x 5 = 25 5 13x13 169 13
6 x 6 = 36 6 14x14 196 14
7 x 7 = 49 7 15x15 225 15
8 x 8 = 64 8 16x16 256 16

Note: Think of these examples: 4 x 4 = 16 and - 4 x - 4 = 16.
Positive numbers have two square roots; a positive (called the principal square root) and a negative. Unless you are asked for the negative square root, you can just give the principal square root.

When exploring squares and square roots with your children, emphasize that each process is the inverse of the other. In other words, one undoes the other.

Writing Square Roots

illustrated example of the square root symbol

Make sure that your children know the square root symbol and that they can find and use it on a calculator.

Calculating and Estimating Square Roots

Most calculators have a square root button that quickly calculates square roots. There are other ways of calculating square roots but they aren't quick. If you need an approximate value for a square root you can use a method like the one below.

Calculating and Estimating Square Roots

Squares and Square Roots Charts

There are two square number charts, both shown above. You will find the printable version of the one on the left here and the 12 to 152 one on the right here.