An exponent tells you how many times a number (or other quantity) is multiplied by itself.
The Oxford English Dictionary defines an exponent as "a symbol denoting the number of times a particular quantity is to be taken as a factor to produce the power indicated."
It is easier to understand exponents with an example. Imagine we wanted to multiply 3 by itself 2 times. We would write 3 with a small 2 written to the right and above it: 32.
- A power is the product of multiplying a number by itself. 32 is equal to 3 multiplied by itself 2 times, which equals 9.
- 3 is called the base. It is the number that is multiplying itself.
- 2 is called the exponent. It tells you how many times the base is multiplying itself.
- The power below has an exponent of 3:
- The power below has an exponent of 4:
- The exponent can also be a letter. The power below has an exponent of n:
Read more about how to find a negative exponent An exponent can be a fraction. A fractional exponent means finding a root of a number. For example, 3½ means the square root of 3:
- 3 to the power of 2.
- The second power of 2.
- 3 to the 2.
- In the special case where the exponent is 2, we can say 3 "squared".
Read more about the laws of exponents If a number or a letter does not have an exponent, imagine it has an invisible exponent of 1:
Lesson Objectives
- Demonstrate an understanding of multiplication with whole numbers
- Demonstrate an understanding of multiplication by powers of 10 (trailing zeros)
- Learn how to write the repeated multiplication of the same whole number using exponents
- Learn how to evaluate an exponential expression
- Learn how to quickly evaluate 10 raised to a whole number exponent
The exponent of a number says how many times to use the number in a multiplication.
In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
In words: 82 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"
Some more examples:Example: 53 = 5 × 5 × 5 = 125
- In words: 53 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"
Example: 24 = 2 × 2 × 2 × 2 = 16
- In words: 24 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"
Exponents make it easier to write and use many multiplications
Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
Note: Exponents are also called Powers or Indices.
You can multiply any number by itself
as many times as you want using exponents.
Try here:
algebra/images/exponent-calc.js
So in general:
| an tells you to multiply a by itself, so there are n of those a's: |
Another Way of Writing It
Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.
Example: 2^4 is the same as 24
Negative Exponents
Negative? What could be the opposite of multiplying? Dividing!
So we divide by the number each time, which is the same as multiplying by 1number
Example: 8-1 = 18 = 0.125
We can continue on like this:
Example: 5-3 = 15 × 15 × 15 = 0.008
But it is often easier to do it this way:
5-3 could also be calculated like:
15 × 5 × 5 = 153 = 1125 = 0.008
Negative? Flip the Positive!
|
That last example showed an easier way to handle negative exponents:
|
More Examples:
| 4-2 | = | 1 / 42 | = | 1/16 = 0.0625 |
| 10-3 | = | 1 / 103 | = | 1/1,000 = 0.001 |
| (-2)-3 | = | 1 / (-2)3 | = | 1/(-8) = -0.125 |
What if the Exponent is 1, or 0?
| 1 | If the exponent is 1, then you just have the number itself (example 91 = 9) | |
| 0 | If the exponent is 0, then you get 1 (example 90 = 1) | |
| But what about 00 ? It could be either 1 or 0, and so people say it is "indeterminate". |
It All Makes Sense
If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern:
| .. etc.. | |||
| 52 | 5 × 5 | 25 | |
| 51 | 5 | 5 | |
| 50 | 1 | 1 | |
| 5-1 | 15 | 0.2 | |
| 5-2 | 15 × 15 | 0.04 | |
| .. etc.. |
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
| With () : | (−2)2 = (−2) × (−2) = 4 |
| Without () : | −22 = −(22) = −(2 × 2) = −4 |
| With () : | (ab)2 = ab × ab |
| Without () : | ab2 = a × (b)2 = a × b × b |
305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864
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