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 Depdendent Variable

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 Dependent Variable

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# Integer Exponents

As we have seen, a positive integer exponent indicates repeated multiplication of a number.

For example, 45 indicates five factors, and each factor is 4.

â€¢ The base, 4, is the repeated factor.

â€¢ The exponent, 5, indicates the number of times the base appears as a factor.

The base can also be a variable, as in x6.

The exponent can also be a variable, as in xn.

Example 1

For each of the following, identify the base and the exponent. Then simplify:

Solution

 a. The base is 3 and the exponent is 4. 34 indicates 4 factors. Each is 3. 34= 3 Â· 3 Â· 3 Â· 3 = 81 b. The base is and the exponent is 2. There are two factors. Each is . c. The base is 2w and the exponent is 3. There are three factors. Each is 2w. (2w)3= (2w) Â· (2w) Â· (2w) = (2 Â· 2 Â· 2) Â· (w Â· w Â· w) = 8w3 In 2w6, the base is w: 2w6 = 2 Â· w Â· w Â· w Â· w Â· w Â· w In (2w)6, the base is 2w: (2w)6 = 2w Â· 2w Â· 2w Â· 2w Â· 2w Â· 2w

Therefore,(2w)3 = 8w3.

d. Because there are no parentheses in the expression 2w6, the base is w, not 2w. The exponent, 6, does not affect the 2. Thus, 2w6 cannot be simplified.

 e. In -24, the base is 2, not -2. The exponent is 4. Thus, there are four factors. Each is 2. -24 = -(2 Â· 2 Â· 2 Â· 2) = -16 Here's a different example. In this case, the base is negative two: (-2)4 = (-2)(-2)(-2)(-2) = +16

Now, letâ€™s look at exponents of 1 and 0, and negative exponents.

â€¢ If an exponent is 1, then the base can be written without an exponent. For example, 51 = 5 and x1 = x.

â€¢ If an exponent is 0, then the value of the expression is 1. For example, 50 = 1 and x0 = 1 (here, x 0).

â€¢ A negative exponent indicates a fraction. For example, . In general, (here, x 0).

It is also true that

For example,

Example 2

Find: 2-1(3x)0y1

 Solution 2-1(3x)0y1 The factor 2-1 can be written as a fraction.The factor (3x)0 can be written as 1.  The factor y1 can be written as y. Simplify.

Thus, the result is .