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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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Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Prove it
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192
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Practice Key Terms 1

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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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