# Laws of Exponents

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Laws of Exponents

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### MATH 1 – Laws of Exponents:

MATH 1 – Laws of Exponents Arnie Dris

### Positive Integer Exponents:

Positive Integer Exponents A 2 = A x A (read as “ A squared equals A times A ”) A 3 = A x A x A (read as “ A cubed”) A 4 = A x A x A x A (read as “ A to the fourth”)

### Positive Integer Exponents (continued):

Positive Integer Exponents (continued) In general: A n = A x A x A x … x A , where there are n factors of A . The number A is called the base , and the positive integer n is called the exponent. In particular, A 1 = A .

### Examples 1:

Examples 1 Evaluate: 2 4 = 2 x 2 x 2 x 2 = 16 (-6) 3 = (-6) x (-6) x (-6) = -216 (3 a ) 2 = (3 a ) x (3 a ) = 9 a 2 Notice the difference between (3 a ) 2 and 3 a 2 : (3 a ) 2 = (3 a ) x (3 a ) = 9 a 2 3 a 2 = 3 x a x a = 3 a 2

### Juxtaposition:

Juxtaposition Henceforth, for simplicity, we shall use juxtaposition to denote multiplication, as in the following: 5(3) 2 = (5)(3)(3) = 45 (2 4 )(2 6 ) = (2)(2)(2)(2)(2)(2)(2)(2)(2)(2) = 2 10 = 1024

### Laws of Exponents 1:

Laws of Exponents 1 This last example: (2 4 )(2 6 ) = (2)(2)(2)(2)(2)(2)(2)(2)(2)(2) = 2 10 = 1024 illustrates a general property: By definition the product a m a n is equal to the product of m a 's and n a 's. This is the product of ( m + n ) a 's. Hence the rule: a m a n = a m + n where m and n are positive integers. This formula could be remembered in words as: Add exponents when multiplying numbers with the same bases.

### Laws of Exponents 2:

Laws of Exponents 2 Now, by the definition of the n th power of a number and the use of the commutative property of multiplication: A n B n = ( AB ) n where n is a positive integer In fact: ( AB ) n = ( AB )( AB )( AB )...( AB ) to n factors = [ (A)(A)(A)...(A) to n factors ] [ (B)(B)(B)...(B) to n factors ] = A n B n Hence, the rule ( AB ) n = A n B n . This formula could be remembered in words as: Multiply bases when multiplying numbers with the same exponents.

### Examples 2:

Examples 2 Evaluate: (2 3 ) 2 = (2 3 )(2 3 ) = 2 3(2) = 2 6 = 64 (3 4 )(3 3 ) = 3 4+3 = 3 7 = 2187

### Laws of Exponents 3:

Laws of Exponents 3 Now, ( a N ) M = ( a N )( a N )( a N )...( a N ) to M factors = a N + N + N +...+ N where there are M N 's = a MN Hence, the rule: ( a N ) M = a MN where M and N are positive integers. This formula could be remembered in words as: To find a power of a power , multiply the exponents .

### Laws of Exponents 4:

Laws of Exponents 4 Likewise, for division, we have: ( A / B ) N = A N / B N provided B ≠ 0

### Examples 3:

Examples 3 Evaluate: (24) 3 /6 3 = (24/6) 3 = 4 3 = 64 3 7 /3 5 = [ (3)(3)(3)(3)(3)(3)(3) ] / [ (3)(3)(3)(3)(3) ] = (3)(3) = 3 2 = 9

### Laws of Exponents 5:

Laws of Exponents 5 This last example: 3 7 /3 5 = [ (3)(3)(3)(3)(3)(3)(3) ] / [ (3)(3)(3)(3)(3) ] = (3)(3) = 3 2 = 9 illustrates the following property: a M / a N = a M - N provided M > N and a ≠ 0 Later on, we shall see that a 0 = 1 and a - N = 1/ a N .

### Seatwork:

Seatwork Evaluate the following: (3 X 2 Y 3 )(4 XY 5 ) (15 X 6 Y 5 )/(3 X 3 Y 4 ) (4 X 2 Y 3 Z 7 ) 3 ((5 X 5 Y 8 )/(3 Z 4 W 2 )) 3 