**Dear Lois:**Could you explain why you add the exponents when you multiply powers of ten? Also why you subtract the exponents when you divide powers of ten?*Cindy*

**Dear Cindy:**I must admit that the methods for multiplying and dividing exponents are some of the strange, but true, rules of mathematics. Why would anyone add when they really meant to multiply or subtract when they wanted to divide? But we do. These rules apply to any kind of exponents, not just powers of ten, so first let's look at exponents in general.

**EXPONENTS**

An exponent simply tells us how many times we are going to multiply a number or variable by itself. For instance, when we see **3 ^{4}**, the

But suppose we have a variable with an exponent, such as **X ^{5}**.

**MULTIPLYING EXPONENTS**

Ready to take the next step? Let's think about multiplying **X ^{5}** by

So you see, when we are multiplying with exponents, adding the exponents is just a shortcut.

**MULTIPLYING POWERS OF 10**

The same principle is true for multiplying powers of ten. **10 ^{3} X 10^{2}** just means to multiply ten by itself 3 times, then 2 times more, or 5 times in all.

**DIVIDING EXPONENTS**

Subtracting exponents when we divide them is also just a short cut. Notice how the **X**'s are stacked up in the problem below. We can arrange them so that we match **X**'s in the numerator with **X**'s in the denominator. Since any number (or variable) divided by itself equals **1**, the matched **X**'s can be eliminated. (Some people call this canceling.) The **X**'s that are left over can be multiplied together (that is if we ever find out what **X** is worth.)

**DIVIDING POWERS OF 10**

Would you like to see that done with powers of ten? Works just the same way. See below.

I hope, Cindy, that this explanation has been of help. Best of luck in your studies. Love, Lois

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