Square numbers
We can represent repeated addition using multiplication, for example:
3+3+3+3=4\times3
In a similar way, we can represent repeated multiplication using powers.
To indicate that a number has been multiplied by itself we write the number being multiplied, called the base, and then indicate the number of times it is multiplied by itself by writing this number, called the power (also known as the index or exponent), above and to the right of the base. Here is an example:
3\times3=3^2
This is read as "three to the power of two" or "three raised to the power of 2."
For now we will just be looking at numbers raised to the power of two, known as square numbers or perfect squares. In the above example 3^2 is often referred to as "three squared".
What does it mean to 'square' a number? Look at the pattern below:
You can see above that each formation of dots forms a square.
| \displaystyle 1^2 | \displaystyle = | \displaystyle 1\times1=1 | |
| \displaystyle 2^2 | \displaystyle = | \displaystyle 2\times2=4 | |
| \displaystyle 3^2 | \displaystyle = | \displaystyle 3\times3=9 | |
| \displaystyle 4^2 | \displaystyle = | \displaystyle 4\times4=16 | and so on. |
Exploration
Use the applet below to explore the first 12 square numbers
The number of the squares is equal to the base squared.
Examples
Example 1
Evaluate 4^{2}+9^{2}.
Perfect squares can be obtained by raising a smaller number to the power of two or multiplying a number by itself.
When writing powers, the number being multiplied is called the base, the number of times it is multiplied by itself is called the power. In the example below 3 is the base and 2 is the power.
3\times3=3^2
The square root
If we are asked to find the square root of a value, we are being asked, "What number multiplied by itself would give this value?"
You might also see the square root symbol written with a number inside it, for example, \sqrt{25}.
This means find the square root of 25.
Examples
Example 2
Evaluate \sqrt{5^2+12^2}.
A perfect square is a number that has a whole number as its square root.
To find the square root of a value, we need to find the number that when multiplied by itself would give the original value.
e.g. \sqrt{25}=5 since 5^2=25.