Exponents look like this:
32
which is a base (3 in this case) raised to a power (2 in this case).
And 32 means 3*3 which, of course, equals 9.
When multiplying exponent numbers with SAME BASE, such as
32 * 34
just ADD the exponents and keep the base. So,
32 * 34 = 36
If the bases are not the same, then you cannot simplify it in that manner.
That is to say
35 * 43 = 35 * 43
bases are NOT the same, you cannot combine.
Now, if there are three numbers with exponents and two have
the same base you can combine, regardless of the order.
35 * 43 * 32 = 37 * 43
One more thing, the rule is "when multiplying, add exponents with the same base". Therefore,
a5 * b3 * a2 = a7 * b3
In many ways, division is the opposite of multiplication.
That rule of thumb works here. When dividing exponent numbers with SAME BASE, such as
35 / 33
just SUBTRACT the exponents and keep the base. So,
35 / 33 = 32 because 5-3=2.
If the bases are not the same, then you cannot simplify it in that manner.
That is to say
35 / 43 = 35 / 43
bases are NOT the same, you cannot combine.
Now, if there are three numbers with exponents and two have
the same base you can combine, regardless of the order.
(35 * 43) / 32 = 33 * 43
One more thing, the rule is "when dividing, subtract exponents with the same base". Therefore,
(a5 * b3) / a2 = a3 * b3
And then there can be more complicated problems involving both multiplying and dividing with several numbers,
some with the same base and some not, such as
34 * 74 * 32 * 82 * 72 = 36 * 76 * 82
and
(34 * 74 * 32 * 82 * 72) / (81 * 33 * 92) = (33 * 81 * 76) / 92
The "zeroth power" rule is simple. Any number to the power of zero is one (1).
30=1 and
1500=1 and
-30=1
Note: Mathmeticians do argue about what 00 (zero to the zeroth power).
I choose to let them worry with that.
One important use of exponents is "scientific notation".
Numbers can be expressed in (at least) two fashions.
One is "standard form" which is the way we are accustomed to seeing numbers,
such as 1037 and another way
is scientific notation, such as
1.037 * 103
And
1.037 * 103 is the same value as 1037
The key to understanding scientific notation is understanding
what happens when a number is multiplied (or divided) by 10, 100, 1000, 10000, etc.
And, by the way, 10, 100, 1000, 10000 are equivalent to
101 , 102 ,103 ,104 , respectively
There is one other fact we need to know.
104 is the same thing as 1/10-4 and furthermore
10-4 is the same thing as 1/104
Stated in a general way, when a number with an exponent is "flipped" from the numerator to
to the denominator (or vice versa) then the sign of the exponent changes.
How could this be? Is 101 somehow equal to 1/10-1?
And as a matter of fact, yes it is.
If this makes no sense to you, just memorize it. Generally speaking,
ab = 1⁄a-b
or maybe more to the point
1⁄1000 = 1⁄103
but
1⁄103 = 10-3
so therefore
1⁄1000 = 10-3
All this becomes important in scientific notation.
In scientific notation, which is a "special way" of representing
a number,
the "form" of the number must be
N.abc * 10"somePower"
where N is a number between 1 and 9. And the entire multiplication
process gives the appropriate number. Geeze, say what?
What is the "scientific notation" form of 837.6? Well, it will
certainly start with 8.376. Now the puzzle is to figure out what this
"somePower" has to be.
8.376 * 10"somePower"
I need to move the decimal point two places to the right for
8.376 to equal 837.6, so therefore "somePower" must equal 2,
because 8.376 * 100 equals 837.6 and 10 to the second power is 100.
So,
837.6 => 8.376 * 102
837.6 is standard form
and 8.376 * 102 is Scientific Notation form
But what is the "scientific notation" form of 0.008376? Well, it will
certainly start with 8.376, because that is a requirement of scientific form.
Now the puzzle is to figure out what this
"somePower" has to be.
8.376 * 10"somePower"
But let us apply "common sense" first. To get the decimal of 8.376
in the correct place to represent 0.008376, we must move the decimal
to the left. That is to say we must DIVIDE by a power of 10.
In fact, 8.376⁄1000 is equal to 0.008376 .
But this is the same as
8.376⁄103
And remember,
1⁄103
is the same as
10-3
and therefore
0.008376 is the same as 8.376*10-3