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# Exponents and Polynomials

## Multiplication with Exponents

Definition of power: For n ∈ { natural numbers}, a > 0 and a ≠ 1,

a n = a Â· a Â· a Â· a Â·...Â· a (n factors of a)

[NOTE: We also say "a to the nth power"

Rule of signs: Remember (-1)(-1) = +1 and (-1)(-1)(-1) = - 1

Generally, count the number of minus signs in a product. Attach "+" for even and "â€“" for odd

NOTE: The exponent affects only the number that it immediately touches so you must always put negative numbers in parentheses ( ).

Example 1:

1. 53 = 5Â·5Â·5 = 125 base = 5, exponent = 3 Say "five to the third power"

2. â€“24 = â€“ 2Â·2Â·2Â·2 = â€“ 16 base = 2, exponent = 4 "negative of two to the fourth power"

3. (- 2)4 = (-2)Â·(-2)Â·(-2)Â·(-2) = 16 base = -2, exponent = 4 "negative two to the fourth power"

4. exponent = 3 "negative two-fifths to the third power"

## Multiplication of "powers with the same base"

Example 2:

1. 24 Â·23 = ( 2Â·2Â·2Â·2 ) Ã—(2Â·2Â·2) = 27 Notice that 4 + 3 = 7

2. (- 2)4Â·(- 2)3= [(-2)Â·(-2)Â·(-2)Â·(-2)] Ã— [(-2)Â·(-2)Â·(-2)] = [16] Ã—[â€“ 8] = â€“ 128 = (- 2)7

1. For base a and m, n ∈ { natural numbers} am Â·an = am + n  → "Multiplication is short-cut addition" Add exponents.

Example 3:

1. 25 Â·24 = 29  = 512

2. x7 Â·x9 = x16

## Powering of "powers with the same base"

For base a and m, n ∈ { natural numbers} (am)n = am Â· n [Powering is short-cut multiplication â€“ multiply exponents.]

Say: "To raise a power to a power, multiply exponents."

Example 4:

1. (24 )3 = (2Â·2Â·2Â·2) Ã—(2Â·2Â·2Â·2) Ã—(2Â·2Â·2Â·2) = 212 Notice that 4 Ã—3 = 12

2. ((- 2)3) 3 = [(-2)Â·(-2)Â·(-2)] Ã—[(-2)Â·(-2)Â·(-2)] Ã—[(-2)Â·(-2)Â·(-2)]= (-2)9 = -29 = - 512 [negative raised to an odd power] Notice that 3 Ã—3 = 9

## Powering of "products with different bases"

For bases a and b, and n ∈ { natural numbers} (aÂ·b)n = an Â·bn = → say â€œThe power of product is the product of the powers.â€

Example 5:

1. (2Â·3)3 = (2Â·3)Ã—(2Â·3)Ã—(2Â·3) = (2Â·2Â·2 ) Ã—(3Â·3Â·3) = (23)(33)

2. (-2Â·x2y3)4 = (-2)4Â·(x2)4(y3)4 = 16Â·(x2Ã—4)(y3Ã—4) = 16 Â·x8 Â·y12

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