## Tuesday, November 29, 2011

### exponents: multiplicative property

It's inconvenient to write exponents in HTML; you have to use the "superscript" command. What's easier is using the circumflex (^) to indicate "raised to the power of." That's what I'll be doing in this post as we cover one interesting property of exponents.

First, some vocab.

The term exponent refers to the tiny number that sits above and to the right of another number or variable. It denotes the number of times that the big number next to it-- called the base-- should be multiplied by itself. If, for example, we have something like this:

x3 (hereafter written as x^3)

--this means we multiply x times x times x (written as x*x*x). The variable x is the base; 3 is the exponent.

What would 34 be? (hereafter 3^4)?

Multiply: 3*3*3*3 = 81.

If two exponential expressions have the same base, you can do interesting things with them.

(x^2)*(x^3) = (x^5) Notice that, when you multiply these two quantities, the powers add! The general rule, then, is

(x^a)*(x^b) = x^(a+b)

Example:

(2^2)*(2^4) = (2^6) = 64

CAUTION: Be careful not to confuse an expression like 2^3 with 2*3!! Students often make this mistake. 2 cubed is 8, but 2 times 3 is 6.

How do we know that multiplying exponential expressions with the same bases means adding exponents? We can work it out the long way.

This equation

(x^2)*(x^3) = (x^5)

can be rewritten as

(x*x)*(x*x*x) = x*x*x*x*x (associative property of multiplication)

...which is x^5!

Remember, though, that if the bases are different, you can't multiply two exponential quantities together and expect to add the exponents. Doesn't work.

OK... more later!

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