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Exponents:
Repeated multiplication of the same number is expressed as exponent.
e.g. 17 x 17 x 17 x 17 x 17 = 175
So, When we multiply 17,5 times, we say 17 to the power 5. Here 17 is the base and 5 is the exponent.
In other word, ab means 'b copies of a, all multiplied together'. So, exponent is just a short hand notation for things multiplied repeatedly.
The exponent of a number says how many times to use the number in a multiplication.
Exponents are also called Powers or Indices.
Exponents make it easier to write and use many multiplications
You can have many divides:
But that can be done an easier way:
More Examples:
If you look at that table, you will see that positive, zero or
negative exponents are really part of the same (fairly simple) pattern.
Check Your Skill: Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10
Repeated multiplication of the same number is expressed as exponent.
e.g. 17 x 17 x 17 x 17 x 17 = 175
So, When we multiply 17,5 times, we say 17 to the power 5. Here 17 is the base and 5 is the exponent.
In other word, ab means 'b copies of a, all multiplied together'. So, exponent is just a short hand notation for things multiplied repeatedly.
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
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"
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"
Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
You can multiply any number by itself as many times as you want using exponents.
Try here:In General
So, in general:
| an tells you to multiply a by itself, so there are n of those a's: |
 |
Other Way of Writing It
Sometimes people use the ^ symbol (just above the 6 on your keyboard), because it is easy to type.
Example: 2^4 is the same as 24
- 2^4 = 2 × 2 × 2 × 2 = 16
Negative Exponents
Negative? What could be the opposite of multiplying?
Dividing!
A negative exponent means how many times to
divide one by the number.
Example: 8-1 = 1 ÷ 8 = 0.125
Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
5-3 could also be calculated like:
1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008
In General
 |
That last example showed an easier way to handle negative exponents:
|
| Negative Exponent | Reciprocal of Positive Exponent | Answer | ||
|---|---|---|---|---|
| 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
My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:| Example: Powers of 5 | |||
|---|---|---|---|
| .. etc.. |  |
||
| 52 | 1 × 5 × 5 | 25 | |
| 51 | 1 × 5 | 5 | |
| 50 | 1 | 1 | |
| 5-1 | 1 ÷ 5 | 0.2 | |
| 5-2 | 1 ÷ 5 ÷ 5 | 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 |
Posted in: Exponent & Logarithms
