| OK, before we go any farther with exponents | ||||||
| there are two special cases we need to talk about. | ||||||
| The first special case is where the exponent is zero. | ||||||
| Like in: | ||||||
|
50 |
||||||
| The value of this is something you probably wouldn't have guessed. | ||||||
| The rules say that ANY NUMBER "to the zero power" equals, | ||||||
| get this, ONE! | ||||||
| Too weird eh? | ||||||
| 20 | = | 1 | ||||
| 60 | = | 1 | ||||
| 321,5870 | = | 1 | ||||
| I know, I know. | ||||||
| EVERYBODY says "but stuff like that should equal zero not one!" | ||||||
| Well what can I tell you, it equals 1 and that's the way it is. | ||||||
| Sometimes things in math come up | ||||||
| that at the time you just have to accept and go on. | ||||||
| This is one of them. | ||||||
| (Later on, after we do a bunch more exponent stuff, | ||||||
| we'll come back to this one and see why it works this way.) | ||||||
| The other special case is when we have exponents | ||||||
| that are negative numbers. | ||||||
| Something like: | ||||||
|
|
||||||
|
4-3 |
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| What does this puppy mean? | ||||||
| Before you even try to guess, I'll tell you. | ||||||
| The negative sign just means the number and the exponent | ||||||
| go in the denominator (bottom part) of a fraction. | ||||||
| And if there is nothing else around a 1 goes in the numerator (top part). | ||||||
| So: | ||||||
|
|
||||||
| STOP! HOLD ON! TIME OUT! | ||||||
| Look at this stuff again. If we have: | ||||||
|
42 |
||||||
| It means: | ||||||
|
42 = 4 × 4 = 16 |
||||||
| If we have: | ||||||
|
40 |
||||||
| It means: | ||||||
|
40 = 1 |
||||||
| If we have: | ||||||
|
4-1 |
||||||
| It means: | ||||||
|
|
||||||
| EVEN WHEN THE EXPONENT IS A NEGATIVE NUMBER, | ||||||
| THE ANSWER WE GET IS POSITIVE | ||||||
| OK, it's a number really close to zero, but it's still positive. | ||||||
| OK. | ||||||
| POP QUIZ TIME. True or false | ||||||
|
4-5,000,000 is greater than 0 |
||||||
|
TRUE! |
||||||
| Hey, it's not much greater than zero, | ||||||
| but even a bit bigger than zero is enough. | ||||||
| Got it? Great, let's go on. | ||||||
| Since we've been talking about negative stuff, what about something like this: | ||||||
|
(-6)5 |
||||||
| This one is no special case or anything. | ||||||
| It's just -6 times itself five times. | ||||||
| It's: | ||||||
|
(-6) × (-6) × (-6) × (-6) × (-6) = -7776 |
||||||
| So (-6)5 = -7776. | ||||||
| You may have figured this next part out already, | ||||||
| but I'll tell you anyway. | ||||||
| If you have a negative number "raised to an even number power" | ||||||
| like: | ||||||
|
(-3)4 = 81 |
||||||
| You get a positive number for an answer. | ||||||
|
BUT |
||||||
| If the exponent is an odd number, like: | ||||||
|
(-3)5 = -243 |
||||||
| You get a negative number for an answer. | ||||||
| Let's put it all together. | ||||||
| How about something like this: | ||||||
|
(-5)-4 |
||||||
| OK, no big thing, just take it step by step. | ||||||
| The exponent has a negative sign so put the whole thing in the denominator: | ||||||
| So: | ||||||
|
|
||||||
| Now forget about the rest of the problem for a minute | ||||||
| and just work on the denominator: | ||||||
|
(-5)4 is just (-5) × (-5) × (-5) × (-5) = 625 |
||||||
|
(the exponent is an even number so the answer is positive) |
||||||
| So putting the problem all together: | ||||||
|
|
||||||
| The exponent in that problem was an even number | ||||||
| so the answer comes out positive. | ||||||
| The minus sign on the exponent | ||||||
| means all the "action" happens in the denominator. | ||||||
| Did you notice that when we raised negative numbers to powers | ||||||
| we always use parenthesis? | ||||||
| We write: | ||||||
|
(-3)4 and not -34 |
||||||
| WHY? | ||||||
| Because in the list of what order you do things, | ||||||
| exponents come before +, -, x, ÷! | ||||||
| So with parenthesis we get: | ||||||
|
(-3)4 = (-3) × (-3) × (-3) × (-3) = 81 |
||||||
| And without parenthesis we get: | ||||||
|
- 34 = - (3 × 3 × 3 × 3) = - 81 |
||||||
| Get it? | ||||||
| We now have a more complete list of the order you do things in. | ||||||
| The order is: | ||||||
|
Parenthesis |
||||||
|
Exponents |
||||||
|
Multiplication and Division |
||||||
|
Addition and Subtraction |
||||||
| One way to remember the order these go in | ||||||
| is to make a sentence from the first letters of the words. | ||||||
| Parenthesis | ||||||
| Exponents | ||||||
| Multiplication | ||||||
| Division | ||||||
| Addition | ||||||
| Subtraction | ||||||
| A famous one of these for this is ... | ||||||
| Please | ||||||
| Excuse | ||||||
| My | ||||||
| Dear | ||||||
| Aunt | ||||||
| Sally | ||||||
| But if you don't like that one you can make up your own. | ||||||
|
copyright 2005 Bruce Kirkpatrick |
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