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 Dependent Variable

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# Exponents and Their Properties

Recall:

if n is a positive integer.

New Stuff:

When you have a product of two terms with the same base, the exponents add together.

Rule (The Product Rule)

For any number a and positive integers m and n , a m a n = a m + n.

• Multiplication of powers leads to addition of the exponents. When you raise something to a power, you are really multiplying it by itself several times. Thus, if you raise a power to a power, you should add the exponent to itself several times. In other words, multiply the powers.

Rule (Power Rule)

For any number a and any positive integers m and n , ( a m ) n .

• Since multiplication leads to addition of exponents, it makes sense that division would lead to subtraction.

Rule (The Quotient Rule)

For any number a ( a 0) and any positive integers m and n for which m > n ,.

The quotient rule allows us to determine what a 0 exponent should mean. Suppose that we allow m = n in the rule above. Then we have (since the top and bottom are the same). But we also have . Thus, we should define a 0 = 1.

Definition ( 0 as an Exponent)

For any real number a , a 0, a 0 = 1.

• That this is a good way to define 0 as an exponent is verified by looking at how well this works with the other rules.

The multiplication rule: If this is a good definition, then a n Â· a 0 should be the same as a n Â· 1. Indeed, it is, since a n Â· a 0 = a n + 0 = a n and a n Â· 1 = a n .

The power rule: The power rule is really just an extension of the multiplication rule, so the definition works with the power rule.

The quotient rule: We made the definition specifically so it works with the quotient rule, so this works too.