Exponents
The exponent of a number says how many times to use the number in a multiplication.
In 8^{2} the "2" says to use 8 twice in a multiplication,
so 8^{2} = 8 × 8 = 64
In words: 8^{2} could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"
Some more examples:Example: 5^{3} = 5 × 5 × 5 = 125
- In words: 5^{3} could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"
Example: 2^{4} = 2 × 2 × 2 × 2 = 16
- In words: 2^{4} 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: 9^{6} 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:
So in general:
a^{n} 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 2^{4}
- 2^4 = 2 × 2 × 2 × 2 = 16
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 \frac{1}{number}
Example: 8^{-1} = \frac{1}{8} = 0.125
We can continue on like this:
Example: 5^{-3} = \frac{1}{5} × \frac{1}{5} × \frac{1}{5} = 0.008
But it is often easier to do it this way:
5^{-3} coan also be calculated like this:
\frac{1}{5 × 5 × 5} = \frac{1}{5^{3}} = \frac{1}{125} = 0.008
Negative? Flip the Positive!
That last example showed an easier way to handle negative exponents:
Examples:
Negative Exponent | Reciprocal of Positive Exponent |
Answer | ||
---|---|---|---|---|
4^{-2} | = | 1 / 4^{2} | = | 1/16 = 0.0625 |
10^{-3} | = | 1 / 10^{3} | = | 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 9^{1} = 9) | |
0 | If the exponent is 0, then you get 1 (example 9^{0} = 1) | |
But what about 0^{0} ? 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:
Example: Powers of 5 | |||
---|---|---|---|
.. etc.. | |||
5^{2} | 5 × 5 | 25 | |
5^{1} | 5 | 5 | |
5^{0} | 1 | 1 | |
5^{-1} | \frac{1}{5} | 0.2 | |
5^{-2} | \frac{1}{5} × \frac{1}{5} | 0.04 | |
.. etc.. |
Be Careful About Grouping
To avoid confusion, use parentheses () in cases like this:
With () : | (−2)^{2} = (−2) × (−2) = 4 |
Without () : | −2^{2} = −(2^{2}) = −(2 × 2) = −4 |
With () : | (ab)^{2} = ab × ab |
Without () : | ab^{2} = a × (b)^{2} = a × b × b |