1.03 Exponents
Ideas
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 the number 10 is the base term and the number 3 is the exponent (or index or power). The expression 10^3 is the same as 10\times10\times10, or the number 10 multiplied 3 times.
We often encounter a power of 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 with an exponent (or power) of 2, can be expressed as 5^2, and can be read as "5 to the power of 2" or "five squared".
A number, e.g. 10 to the power of 3, can be expressed as 10^3, and can be read as "ten cubed". A power of 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, can be read as "3 to the power of 4", and means that the base number is multiplied by itself the number of times shown in the power.
3^4=3\times3\times3\times3
To evaluate or simplify the above exponential expression, the only step we need to take is completing the multiplication.
| \displaystyle 3^4 | \displaystyle = | \displaystyle 3\times3\times3\times3 | |
| \displaystyle = | \displaystyle 81 | Simplify the multiplication |
Therefore, we can say that 3^{4} = 81.
Exploration
The following demonstration illustrates more of this notation. Try varying the bases and exponents (by moving the sliders) to see how the numbers change.
The index is equal to the number of times the base is multiplied by itself.
Examples
Example 1
State the base for the expression 3^2.
Example 2
Identify the power for the expression 4^6.
Example 3
Write the following in expanded form: 7^5 \times 6^4
An exponent (or power) notes how many times a base is being multiplied by itself.
A base to the power of any other number means that the base number is multiplied by itself the number of times shown in the power.