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Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

  1. ( −3 t ) 2 ( −3 t ) 8
  2. f 47 f 49 f
  3. 2 k 4 5 k 7
  1. 1 ( −3 t ) 6
  2. 1 f 3
  3. 2 5 k 3
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Using the product and quotient rules

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

  1. b 2 b −8
  2. ( x ) 5 ( x ) −5
  3. −7 z ( −7 z ) 5
  1. b 2 b −8 = b 2 8 = b −6 = 1 b 6
  2. ( x ) 5 ( x ) −5 = ( x ) 5 5 = ( x ) 0 = 1
  3. −7 z ( −7 z ) 5 = ( −7 z ) 1 ( −7 z ) 5 = ( −7 z ) 1 5 = ( −7 z ) −4 = 1 ( −7 z ) 4
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Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

  1. t −11 t 6
  2. 25 12 25 13
  1. t −5 = 1 t 5
  2. 1 25
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Finding the power of a product

To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider ( p q ) 3 . We begin by using the associative and commutative properties of multiplication to regroup the factors.

( p q ) 3 = ( p q ) ( p q ) ( p q ) 3  factors = p q p q p q = p p p 3  factors q q q 3  factors = p 3 q 3

In other words, ( p q ) 3 = p 3 q 3 .

The power of a product rule of exponents

For any real numbers a and b and any integer n , the power of a product rule of exponents states that

( a b ) n = a n b n

Using the power of a product rule

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

  1. ( a b 2 ) 3
  2. ( 2 t ) 15
  3. ( −2 w 3 ) 3
  4. 1 ( −7 z ) 4
  5. ( e −2 f 2 ) 7

Use the product and quotient rules and the new definitions to simplify each expression.

  1. ( a b 2 ) 3 = ( a ) 3 ( b 2 ) 3 = a 1 3 b 2 3 = a 3 b 6
  2. ( 2 t ) 15 = ( 2 ) 15 ( t ) 15 = 2 15 t 15 = 32 , 768 t 15
  3. ( −2 w 3 ) 3 = ( −2 ) 3 ( w 3 ) 3 = −8 w 3 3 = −8 w 9
  4. 1 ( −7 z ) 4 = 1 ( −7 ) 4 ( z ) 4 = 1 2 , 401 z 4
  5. ( e −2 f 2 ) 7 = ( e −2 ) 7 ( f 2 ) 7 = e −2 7 f 2 7 = e −14 f 14 = f 14 e 14
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Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

  1. ( g 2 h 3 ) 5
  2. ( 5 t ) 3
  3. ( −3 y 5 ) 3
  4. 1 ( a 6 b 7 ) 3
  5. ( r 3 s −2 ) 4
  1. g 10 h 15
  2. 125 t 3
  3. −27 y 15
  4. 1 a 18 b 21
  5. r 12 s 8
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Finding the power of a quotient

To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

( e −2 f 2 ) 7 = f 14 e 14

Let’s rewrite the original problem differently and look at the result.

( e −2 f 2 ) 7 = ( f 2 e 2 ) 7 = f 14 e 14

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

( e 2 f 2 ) 7 = ( f 2 e 2 ) 7 = ( f 2 ) 7 ( e 2 ) 7 = f 2 7 e 2 7 = f 14 e 14

The power of a quotient rule of exponents

For any real numbers a and b and any integer n , the power of a quotient rule of exponents states that

( a b ) n = a n b n

Using the power of a quotient rule

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

  1. ( 4 z 11 ) 3
  2. ( p q 3 ) 6
  3. ( −1 t 2 ) 27
  4. ( j 3 k −2 ) 4
  5. ( m −2 n −2 ) 3
  1. ( 4 z 11 ) 3 = ( 4 ) 3 ( z 11 ) 3 = 64 z 11 3 = 64 z 33
  2. ( p q 3 ) 6 = ( p ) 6 ( q 3 ) 6 = p 1 6 q 3 6 = p 6 q 18
  3. ( −1 t 2 ) 27 = ( −1 ) 27 ( t 2 ) 27 = −1 t 2 27 = −1 t 54 = 1 t 54
  4. ( j 3 k −2 ) 4 = ( j 3 k 2 ) 4 = ( j 3 ) 4 ( k 2 ) 4 = j 3 4 k 2 4 = j 12 k 8
  5. ( m −2 n −2 ) 3 = ( 1 m 2 n 2 ) 3 = ( 1 ) 3 ( m 2 n 2 ) 3 = 1 ( m 2 ) 3 ( n 2 ) 3 = 1 m 2 3 n 2 3 = 1 m 6 n 6
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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