| Check this out: | ||
|
(42)3 |
||
| What do you think this one means? | ||
| Well, just take it one step at a time: | ||
|
(42)3 = 42 × 42 × 42 |
||
| And we know that 4 2 = 4 x 4, so: | ||
|
(42)3 = 42 × 42 × 42 = 4 × 4 × 4 × 4 × 4 × 4 = 46 = 4,096 |
||
| Let's try something else. WATCH CLOSELY: | ||
|
52 + 3 = 52 × 53 NOT 52 + 53 |
||
| Here's why: | ||
|
52 + 3 = 55 = 5 × 5 × 5 × 5 × 5 |
||
| Now let's do some grouping with parenthesis: | ||
|
55 = (5 × 5) × (5 × 5 × 5) |
||
|
(Since all of the signs are "x" we don't change anything by grouping) |
||
| The next step is to write each of these groups as a number with an exponent: | ||
|
55 = (5 × 5) × (5 × 5 × 5) = 52 × 53 |
||
| And there you go! | ||
|
|
||
| OK, How about this: | ||
|
23 × 33 |
||
| Can we combine this one? | ||
| Let's see. First, break it down: | ||
|
23 × 33 |
||
|
2 × 2 × 2 × 3 × 3 × 3 |
||
| Now mix up the numbers a bit: | ||
|
2 × 3 × 2 × 3 × 2 × 3 |
||
| Next, put in some parenthesis in special places: | ||
|
(2 × 3) × (2 × 3) × (2 × 3) |
||
| Multiply the stuff inside each parenthesis: | ||
|
6 × 6 × 6 |
||
| And write this as a number and an exponent: | ||
|
63 |
||
| OK, Big Deal, So What, Who Cares? | ||
| Here's the deal. | ||
| We started with: | ||
|
23 × 33 |
||
| And ended with: | ||
|
63 |
||
| That means: | ||
| IF THE EXPONENTS ARE THE SAME | ||
| IN A MULTIPLICATION PROBLEM, | ||
| YOU CAN JUST MULTIPLY THE NUMBERS UNDER THEM! | ||
|
23 × 33 = (2 × 3)3 = 63 |
||
| Examples: | ||
|
75 × 45 = (7 × 4)5 = 285 |
||
|
26 × 46 = (2 × 4)6 = 86 |
||
|
3-2 × 4-2 = (3 × 4)-2 = 12-2 |
||
| What happens with something like: | ||
|
35 + 75 |
||
| Can we leave the exponents in place and add this thing together somehow? | ||
|
NO! |
||
| If you read the chapters up to here, you probably remember problems like: | ||
|
30 = 1 |
||
|
|
||
| Back then, I said that you just had to accept that as so. | ||
| I said that sometime later I would show you why it works that way. | ||
| IT'S TIME! | ||
| At the start of this chapter we had stuff like this: | ||
|
52 × 53 = 5(2 + 3) = 55 |
||
| Check out this one: | ||
|
53 × 5-3 |
||
| We can deal with this one in two ways. The first way is: | ||
|
53 × 5-3 = 5(3 - 3) = 50 = 1 |
||
| Oh yeah? So what? That doesn't prove ANYTHING! | ||
| That's true, but now look at the second way we can do this one: | ||
|
|
||
| Now put a denominator of 1 under the 5 3 and we have: | ||
|
|
||
| Now multiply and simplify: | ||
|
|
||
| And there it is! | ||
|
53 × 5-3 = 50 = 1 |
||
|
copyright 2005 Bruce Kirkpatrick |
