1.01 Exponent notation and perfect squares
Exponent notation
An exponent (or power) is a small number placed in the upper right hand corner of another number to note how many times a base is being multiplied by itself.
For example, in the expression $10^3$103 the number $10$10 is the base term and the number $3$3 is the exponent (or index or power). The expression $10^3$103 is the same as $10\times10\times10$10×10×10, or the number $10$10 multiplied $3$3 times.
Think of the base as that being closest to the ground, and the exponent (or power) is above.
We often encounter a power of $2$2 when measuring area. Consider the area of a square, for example, which is given by side length times side length. A number, e.g. $5$5 with an exponent (or power) of $2$2, can be expressed as $5^2$52, and can be read as "$5$5 to the power of $2$2" or "five squared".
A number, e.g. $10$10 to the power of $3$3, can be expressed as $10^3$103, and can be read as "ten cubed". A power of $3$3 is involved in calculations like measuring the volume of a cube.
A base to the power of any other number, e.g. $3^4$34, can be read as "three to the power of four", and means that the base number is multiplied by itself the number of times shown in the power.
$3^4=3\times3\times3\times3$34=3×3×3×3
The following demonstration illustrates more of this notation. Try varying the bases and exponents (by moving the sliders) to see how the numbers change.
Worked example
Question 1
Identify the base and exponent for the expression $2^5$25 and then rewrite in expanded form.
Think: The base is the number being multiplied by itself and is the "normal" sized number. The exponent is the number of times we will be multiplying and is the "superscript" number up top.
Do: $2$2 is the base and $5$5 is the exponent, so we will multiply $2$2 by itself $5$5 times.
$2^5=2\times2\times2\times2\times2$25=2×2×2×2×2
Practice questions
Question 2
State the base for the expression $3^2$32.
Question 3
Identify the power for the expression $4^6$46.
$6$6
A$4$4
B
Question 4
Perfect squares
Let's focus our attention on numbers raised to the power of two, known as square numbers or perfect squares. For example, the expression $3^2$32 is often referred to as "three squared".
What does it mean to 'square' a number? Look at the pattern below!
| $1^2$12 | $2^2$22 | $3^2$32 | $4^2$42 | $5^2$52 |
|---|---|---|---|---|
 |
 |
 |
 |
 |
| 1 | 4 | 9 | 16 | 25 |
You can see above that each formation of dots forms a square.
| $1^2$12 | $=$= | $1\times1$1×1 | $=$= | $1$1 | |
| $2^2$22 | $=$= | $2\times2$2×2 | $=$= | $4$4 | |
| $3^2$32 | $=$= | $3\times3$3×3 | $=$= | $9$9 | |
| $4^2$42 | $=$= | $4\times4$4×4 | $=$= | $16$16 | and so on. |
Exploration
Use the applet below to explore the first $20$20 perfect squares. Consider the questions below as you explore.
- Which of these perfect squares are you already familiar with?
- How can you create a definition for a perfect square, using tiles?
- Where might you see these numbers come up outside of math class?
- What is the relationship between the area (number of tiles) and side length?
- Why do you think they call multiplying a number by itself as squaring a number?
Worked example
Question 5
Evaluate $7^2$72.
Think: Squaring $7$7 is multiplying $7$7 by itself.
Do:
| $7^2$72 |
Given |
|
| $=$= | $7\times7$7×7 |
Definition of squaring (optional) |
| $=$= | $49$49 |
Simplifying |
Practice question
QUESTION 6
Evaluate $9^2$92