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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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cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
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how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
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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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