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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

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
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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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