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REVIEW OF MULTIPLICATION

So again, what is multiplication?

Multiplication is addition of a single number multiple times.

So, 13*13 is really: 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

Another example would be 4 * 20:

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4

So in other words, multiplication is an easier way to write the addition of a single number multiple times.

So what about when you want to multiply a single number multiple times?

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Definition of Exponents

So according to google, an exponent is:

“A quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression.”

Or, in English, it’s an easier way to write the multiplication of a single number, multiple times. The way we write this is as so:

Let’s say, we want to write the expression: 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3

as an exponent instead. First thing we do is find the base.

The base of the exponent is the number or expression that is being multiplied together by itself multiple times.

So in this case, our base is 3.

Next we count how many 3’s we have to find out what our exponent is.

So in our example:

3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

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THE ADDITION, MULTIPLICATION, EXPONENTS LEVELS

So, we know multiplication is just addition of the same number multiple times. We also know that if we are given an x-y table such as:

X	Y
0	0
1	3
2	6

The equation to this x-y table is going to be y = 3x. We know this because we add 3 to the y values to get the next value.


+3
+3

Exponent

Multiplication

Addition

Using the levels graph below, if we start at addition for the x-y table, then we will move up one level to multiplication for the equation.

Similarly, if we see that the change in Y in an X-Y table is multiplication, then we move from multiplication, up one level to Exponent for the equation.

So if we have:

X	Y
0	1
1	5
2	25


×5
×5

So, again, using the levels graph to the left, if we start at multiplication for the x-y table, then we will move up one level to exponent for the equation.

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A FEW THINGS TO NOTE

When you see a recursive pattern in either a list of numbers (a sequence) or in an x-y table, and you can tell immediately that it is being multiplied (or divided) by a single number, whatever that number is will be the base of the exponent.

Ex.

X	Y
0	1
1	2
2	4
3	8
4	16
5	32


×2
×2
×2
×2
×2

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ANOTHER EXAMPLE:

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HOW DO WE KNOW IF IT IS BEING ADDED OR MULTIPLIED?

So sometimes it’s going to be a recursive multiplication problem.

Sometimes it’s going to be a recursive addition problem.

So how do we tell which is which?

Sometimes it will be pretty obvious that we are not adding, but instead multiplying. Sometimes it will be pretty clear that you are adding and not multiplying.

Basically in order to find out, you need to try both.

So, in this case we use the slope-formula.

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Slope formula

x	Y
0	0
1	13
2	26
3	39

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When slope formula doesn’t work.

Now let’s say instead we are given something like:

X	Y
0	1
1	15
2	225
3	3375
4	50625
5	759375

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The power of 0

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Another proof for anything to the zero power.

X	Y
1	2
2	4
3	8
4	16

Again, we can see that we are multiplying by 2 each time.


×2
×2
×2

However, if every time we go down the x-y table we multiply by 2, then what would happen if instead we went up?

X	Y
0
1	2	÷2
2	4	÷2
3	8	÷2

So we can see that we will divide by 2 each time. So, if we continue dividing by 2, what would happen if we divide 2 by 2?