unit 6 homework 3 dividing monomials

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Unit 6 Homework 3: Dividing Monomials

9th - 12th grade, mathematics.

10 questions

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Simplify the following monomials:   k 6 k 6 \frac{k^6}{k^6} k 6 k 6 ​    

  k 12 k^{12} k 1 2    

  k 6 k^6 k 6    

Simplify the following monomials:   m n 3 n 2 \frac{mn^3}{n^2} n 2 m n 3 ​    

  m n mn m n    

  1 2 m n \frac{1}{2}mn 2 1 ​ m n    

  m − 1 n m^{-1}n m − 1 n    

Simplify the following monomials:   8 x 5 y 4 4 x 2 y 2 \frac{8x^5y^4}{4x^2y^2} 4 x 2 y 2 8 x 5 y 4 ​    

  2 x 3 y 2 2x^3y^2 2 x 3 y 2    

  4 x 3 y 2 4x^3y^2 4 x 3 y 2    

  12 x 7 y 6 12x^7y^6 1 2 x 7 y 6    

Simplify the following monomials:   − 20 x 3 y 2 − 5 x 3 y \frac{-20x^3y^2}{-5x^3y} − 5 x 3 y − 2 0 x 3 y 2 ​    

  4 y 4y 4 y    

  − 25 x 3 y -25x^3y − 2 5 x 3 y    

  4 x y 4xy 4 x y    

Simplify the following monomials:   21 m 8 n 5 27 m 5 n 4 \frac{21m^8n^5}{27m^5n^4} 2 7 m 5 n 4 2 1 m 8 n 5 ​    

  7 9 m 3 n \frac{7}{9}m^3n 9 7 ​ m 3 n    

  − 6 m 3 n -6m^3n − 6 m 3 n    

  9 7 m 3 n \frac{9}{7}m^3n 7 9 ​ m 3 n    

Simplify the following monomials:   7 p 2 q 2 14 p 2 q 2 \frac{7p^2q^2}{14p^2q^2} 1 4 p 2 q 2 7 p 2 q 2 ​    

  1 2 \frac{1}{2} 2 1 ​    

  − 7 -7 − 7    

  2 2 2    

Simplify the following monomials:   ( 2 a 3 b 4 ) 3 ( 2 a b 2 ) 5 \frac{\left(2a^3b^4\right)^3}{\left(2ab^2\right)^5} ( 2 a b 2 ) 5 ( 2 a 3 b 4 ) 3 ​    

  1 4 a 4 b 2 \frac{1}{4}a^4b^2 4 1 ​ a 4 b 2    

  32 a 5 b 10 32a^5b^{10} 3 2 a 5 b 1 0    

  8 a 9 b 12 8a^9b^{12} 8 a 9 b 1 2    

Simplify the following monomials:   ( 4 x 7 6 x ) 2 \left(\frac{4x^7}{6x}\right)^2 ( 6 x 4 x 7 ​ ) 2    

  4 9 x 12 \frac{4}{9}x^{12} 9 4 ​ x 1 2    

  16 x 14 16x^{14} 1 6 x 1 4    

  36 x 2 36x^2 3 6 x 2    

Simplify the following monomials:   ( 5 x 6 y 2 10 x 4 y ) 2 \left(\frac{5x^6y^2}{10x^4y}\right)^2 ( 1 0 x 4 y 5 x 6 y 2 ​ ) 2    

  1 4 x 4 y 2 \frac{1}{4}x^4y^2 4 1 ​ x 4 y 2    

  25 x 12 y 4 25x^{12}y^4 2 5 x 1 2 y 4    

  100 x 8 y 2 100x^8y^2 1 0 0 x 8 y 2    

Simplify the following monomials:   ( 2 x 3 ) 2 ( 3 y 4 ) 3 12 x 4 y 5 − 4 x 2 y 7 \frac{\left(2x^3\right)^2\left(3y^4\right)^3}{12x^4y^5}-4x^2y^7 1 2 x 4 y 5 ( 2 x 3 ) 2 ( 3 y 4 ) 3 ​ − 4 x 2 y 7    

  5 x 2 y 7 5x^2y^7 5 x 2 y 7    

  9 x 2 y 7 9x^2y^7 9 x 2 y 7    

  − 4 x 2 y 7 -4x^2y^7 − 4 x 2 y 7    

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6.5 Divide Monomials

Learning objectives.

By the end of this section, you will be able to:

• Simplify expressions using the Quotient Property for Exponents
• Simplify expressions with zero exponents
• Simplify expressions using the quotient to a Power Property
• Simplify expressions by applying several properties
• Divide monomials

Be Prepared 6.9

Before you get started, take this readiness quiz.

Simplify: 8 24 . 8 24 . If you missed this problem, review Example 1.65 .

Be Prepared 6.10

Simplify: ( 2 m 3 ) 5 . ( 2 m 3 ) 5 . If you missed this problem, review Example 6.23 .

Be Prepared 6.11

Simplify: 12 x 12 y . 12 x 12 y . If you missed this problem, review Example 1.67 .

Simplify Expressions Using the Quotient Property for Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.

Summary of Exponent Properties for Multiplication

If a and b a and b are real numbers, and m and n m and n are whole numbers, then

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

Equivalent Fractions Property

If a , b , and c a , b , and c are whole numbers where b ≠ 0 , c ≠ 0 b ≠ 0 , c ≠ 0 ,

As before, we’ll try to discover a property by looking at some examples.

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

This leads to the Quotient Property for Exponents .

Quotient Property for Exponents

If a a is a real number, a ≠ 0 a ≠ 0 , and m and n m and n are whole numbers, then

A couple of examples with numbers may help to verify this property.

Example 6.59

Simplify: ⓐ x 9 x 7 x 9 x 7 ⓑ 3 10 3 2 . 3 10 3 2 .

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

Try It 6.117

Simplify: ⓐ x 15 x 10 x 15 x 10 ⓑ 6 14 6 5 . 6 14 6 5 .

Try It 6.118

Simplify: ⓐ y 43 y 37 y 43 y 37 ⓑ 10 15 10 7 . 10 15 10 7 .

Example 6.60

Simplify: ⓐ b 8 b 12 b 8 b 12 ⓑ 7 3 7 5 . 7 3 7 5 .

Try It 6.119

Simplify: ⓐ x 18 x 22 x 18 x 22 ⓑ 12 15 12 30 . 12 15 12 30 .

Try It 6.120

Simplify: ⓐ m 7 m 15 m 7 m 15 ⓑ 9 8 9 19 . 9 8 9 19 .

Notice the difference in the two previous examples:

• If we start with more factors in the numerator, we will end up with factors in the numerator.
• If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

Example 6.61

Simplify: ⓐ a 5 a 9 a 5 a 9 ⓑ x 11 x 7 . x 11 x 7 .

Try It 6.121

Simplify: ⓐ b 19 b 11 b 19 b 11 ⓑ z 5 z 11 . z 5 z 11 .

Try It 6.122

Simplify: ⓐ p 9 p 17 p 9 p 17 ⓑ w 13 w 9 . w 13 w 9 .

Simplify Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like a m a m a m a m . From your earlier work with fractions, you know that:

In words, a number divided by itself is 1. So, x x = 1 x x = 1 , for any x ( x ≠ 0 ) x ( x ≠ 0 ) , since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify a m a n a m a n when m > n m > n and when n < m n < m by subtracting exponents. What if m = n m = n ?

Consider 8 8 8 8 , which we know is 1.

Now we will simplify a m a m a m a m in two ways to lead us to the definition of the zero exponent. In general, for a ≠ 0 a ≠ 0 :

We see a m a m a m a m simplifies to a 0 a 0 and to 1. So a 0 = 1 a 0 = 1 .

Zero Exponent

If a a is a non-zero number, then a 0 = 1 a 0 = 1 .

Any nonzero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example 6.62

Simplify: ⓐ 9 0 9 0 ⓑ n 0 . n 0 .

The definition says any non-zero number raised to the zero power is 1.

Try It 6.123

Simplify: ⓐ 15 0 15 0 ⓑ m 0 . m 0 .

Try It 6.124

Simplify: ⓐ k 0 k 0 ⓑ 29 0 . 29 0 .

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let’s look at ( 2 x ) 0 ( 2 x ) 0 . We can use the product to a power rule to rewrite this expression.

This tells us that any nonzero expression raised to the zero power is one.

Example 6.63

Simplify: ⓐ ( 5 b ) 0 ( 5 b ) 0 ⓑ ( −4 a 2 b ) 0 . ( −4 a 2 b ) 0 .

Try It 6.125

Simplify: ⓐ ( 11 z ) 0 ( 11 z ) 0 ⓑ ( −11 p q 3 ) 0 . ( −11 p q 3 ) 0 .

Try It 6.126

Simplify: ⓐ ( −6 d ) 0 ( −6 d ) 0 ⓑ ( −8 m 2 n 3 ) 0 . ( −8 m 2 n 3 ) 0 .

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

Notice that the exponent applies to both the numerator and the denominator.

This leads to the Quotient to a Power Property for Exponents .

Quotient to a Power Property for Exponents

If a a and b b are real numbers, b ≠ 0 b ≠ 0 , and m m is a counting number, then

To raise a fraction to a power, raise the numerator and denominator to that power.

An example with numbers may help you understand this property:

Example 6.64

Simplify: ⓐ ( 3 7 ) 2 ( 3 7 ) 2 ⓑ ( b 3 ) 4 ( b 3 ) 4 ⓒ ( k j ) 3 . ( k j ) 3 .

Try It 6.127

Simplify: ⓐ ( 5 8 ) 2 ( 5 8 ) 2 ⓑ ( p 10 ) 4 ( p 10 ) 4 ⓒ ( m n ) 7 . ( m n ) 7 .

Try It 6.128

Simplify: ⓐ ( 1 3 ) 3 ( 1 3 ) 3 ⓑ ( −2 q ) 3 ( −2 q ) 3 ⓒ ( w x ) 4 . ( w x ) 4 .

Simplify Expressions by Applying Several Properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of Exponent Properties

Example 6.65.

Simplify: ( y 4 ) 2 y 6 . ( y 4 ) 2 y 6 .

Try It 6.129

Simplify: ( m 5 ) 4 m 7 . ( m 5 ) 4 m 7 .

Try It 6.130

Simplify: ( k 2 ) 6 k 7 . ( k 2 ) 6 k 7 .

Example 6.66

Simplify: b 12 ( b 2 ) 6 . b 12 ( b 2 ) 6 .

Try It 6.131

Simplify: n 12 ( n 3 ) 4 . n 12 ( n 3 ) 4 .

Try It 6.132

Simplify: x 15 ( x 3 ) 5 . x 15 ( x 3 ) 5 .

Example 6.67

Simplify: ( y 9 y 4 ) 2 . ( y 9 y 4 ) 2 .

Try It 6.133

Simplify: ( r 5 r 3 ) 4 . ( r 5 r 3 ) 4 .

Try It 6.134

Simplify: ( v 6 v 4 ) 3 . ( v 6 v 4 ) 3 .

Example 6.68

Simplify: ( j 2 k 3 ) 4 . ( j 2 k 3 ) 4 .

Here we cannot simplify inside the parentheses first, since the bases are not the same.

Try It 6.135

Simplify: ( a 3 b 2 ) 4 . ( a 3 b 2 ) 4 .

Try It 6.136

Simplify: ( q 7 r 5 ) 3 . ( q 7 r 5 ) 3 .

Example 6.69

Simplify: ( 2 m 2 5 n ) 4 . ( 2 m 2 5 n ) 4 .

Try It 6.137

Simplify: ( 7 x 3 9 y ) 2 . ( 7 x 3 9 y ) 2 .

Try It 6.138

Simplify: ( 3 x 4 7 y ) 2 . ( 3 x 4 7 y ) 2 .

Example 6.70

Simplify: ( x 3 ) 4 ( x 2 ) 5 ( x 6 ) 5 . ( x 3 ) 4 ( x 2 ) 5 ( x 6 ) 5 .

Try It 6.139

Simplify: ( a 2 ) 3 ( a 2 ) 4 ( a 4 ) 5 . ( a 2 ) 3 ( a 2 ) 4 ( a 4 ) 5 .

Try It 6.140

Simplify: ( p 3 ) 4 ( p 5 ) 3 ( p 7 ) 6 . ( p 3 ) 4 ( p 5 ) 3 ( p 7 ) 6 .

Example 6.71

Simplify: ( 10 p 3 ) 2 ( 5 p ) 3 ( 2 p 5 ) 4 . ( 10 p 3 ) 2 ( 5 p ) 3 ( 2 p 5 ) 4 .

Try It 6.141

Simplify: ( 3 r 3 ) 2 ( r 3 ) 7 ( r 3 ) 3 . ( 3 r 3 ) 2 ( r 3 ) 7 ( r 3 ) 3 .

Try It 6.142

Simplify: ( 2 x 4 ) 5 ( 4 x 3 ) 2 ( x 3 ) 5 . ( 2 x 4 ) 5 ( 4 x 3 ) 2 ( x 3 ) 5 .

Divide Monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Example 6.72

Find the quotient: 56 x 7 ÷ 8 x 3 . 56 x 7 ÷ 8 x 3 .

Try It 6.143

Find the quotient: 42 y 9 ÷ 6 y 3 . 42 y 9 ÷ 6 y 3 .

Try It 6.144

Find the quotient: 48 z 8 ÷ 8 z 2 . 48 z 8 ÷ 8 z 2 .

Example 6.73

Find the quotient: 45 a 2 b 3 −5 a b 5 . 45 a 2 b 3 −5 a b 5 .

Try It 6.145

Find the quotient: −72 a 7 b 3 8 a 12 b 4 . −72 a 7 b 3 8 a 12 b 4 .

Try It 6.146

Find the quotient: −63 c 8 d 3 7 c 12 d 2 . −63 c 8 d 3 7 c 12 d 2 .

Example 6.74

Find the quotient: 24 a 5 b 3 48 a b 4 . 24 a 5 b 3 48 a b 4 .

Try It 6.147

Find the quotient: 16 a 7 b 6 24 a b 8 . 16 a 7 b 6 24 a b 8 .

Try It 6.148

Find the quotient: 27 p 4 q 7 −45 p 12 q . 27 p 4 q 7 −45 p 12 q .

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example 6.75

Find the quotient: 14 x 7 y 12 21 x 11 y 6 . 14 x 7 y 12 21 x 11 y 6 .

Be very careful to simplify 14 21 14 21 by dividing out a common factor, and to simplify the variables by subtracting their exponents.

Try It 6.149

Find the quotient: 28 x 5 y 14 49 x 9 y 12 . 28 x 5 y 14 49 x 9 y 12 .

Try It 6.150

Find the quotient: 30 m 5 n 11 48 m 10 n 14 . 30 m 5 n 11 48 m 10 n 14 .

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

Example 6.76

Find the quotient: ( 6 x 2 y 3 ) ( 5 x 3 y 2 ) ( 3 x 4 y 5 ) . ( 6 x 2 y 3 ) ( 5 x 3 y 2 ) ( 3 x 4 y 5 ) .

Try It 6.151

Find the quotient: ( 6 a 4 b 5 ) ( 4 a 2 b 5 ) 12 a 5 b 8 . ( 6 a 4 b 5 ) ( 4 a 2 b 5 ) 12 a 5 b 8 .

Try It 6.152

Find the quotient: ( −12 x 6 y 9 ) ( −4 x 5 y 8 ) −12 x 10 y 12 . ( −12 x 6 y 9 ) ( −4 x 5 y 8 ) −12 x 10 y 12 .

Access these online resources for additional instruction and practice with dividing monomials:

• Rational Expressions
• Dividing Monomials
• Dividing Monomials 2

Section 6.5 Exercises

Practice makes perfect.

In the following exercises, simplify.

ⓐ x 18 x 3 x 18 x 3 ⓑ 5 12 5 3 5 12 5 3

ⓐ y 20 y 10 y 20 y 10 ⓑ 7 16 7 2 7 16 7 2

ⓐ p 21 p 7 p 21 p 7 ⓑ 4 16 4 4 4 16 4 4

ⓐ u 24 u 3 u 24 u 3 ⓑ 9 15 9 5 9 15 9 5

ⓐ q 18 q 36 q 18 q 36 ⓑ 10 2 10 3 10 2 10 3

ⓐ t 10 t 40 t 10 t 40 ⓑ 8 3 8 5 8 3 8 5

ⓐ b b 9 b b 9 ⓑ 4 4 6 4 4 6

ⓐ x x 7 x x 7 ⓑ 10 10 3 10 10 3

Simplify Expressions with Zero Exponents

ⓐ 20 0 20 0 ⓑ b 0 b 0

ⓐ 13 0 13 0 ⓑ k 0 k 0

ⓐ − 27 0 − 27 0 ⓑ − ( 27 0 ) − ( 27 0 )

ⓐ − 15 0 − 15 0 ⓑ − ( 15 0 ) − ( 15 0 )

ⓐ ( 25 x ) 0 ( 25 x ) 0 ⓑ 25 x 0 25 x 0

ⓐ ( 6 y ) 0 ( 6 y ) 0 ⓑ 6 y 0 6 y 0

ⓐ ( 12 x ) 0 ( 12 x ) 0 ⓑ ( −56 p 4 q 3 ) 0 ( −56 p 4 q 3 ) 0

ⓐ 7 y 0 7 y 0 ( 17 y ) 0 ( 17 y ) 0 ⓑ ( −93 c 7 d 15 ) 0 ( −93 c 7 d 15 ) 0

ⓐ 12 n 0 − 18 m 0 12 n 0 − 18 m 0 ⓑ ( 12 n ) 0 − ( 18 m ) 0 ( 12 n ) 0 − ( 18 m ) 0

ⓐ 15 r 0 − 22 s 0 15 r 0 − 22 s 0 ⓑ ( 15 r ) 0 − ( 22 s ) 0 ( 15 r ) 0 − ( 22 s ) 0

ⓐ ( 3 4 ) 3 ( 3 4 ) 3 ⓑ ( p 2 ) 5 ( p 2 ) 5 ⓒ ( x y ) 6 ( x y ) 6

ⓐ ( 2 5 ) 2 ( 2 5 ) 2 ⓑ ( x 3 ) 4 ( x 3 ) 4 ⓒ ( a b ) 5 ( a b ) 5

ⓐ ( a 3 b ) 4 ( a 3 b ) 4 ⓑ ( 5 4 m ) 2 ( 5 4 m ) 2

ⓐ ( x 2 y ) 3 ( x 2 y ) 3 ⓑ ( 10 3 q ) 4 ( 10 3 q ) 4

( a 2 ) 3 a 4 ( a 2 ) 3 a 4

( p 3 ) 4 p 5 ( p 3 ) 4 p 5

( y 3 ) 4 y 10 ( y 3 ) 4 y 10

( x 4 ) 5 x 15 ( x 4 ) 5 x 15

u 6 ( u 3 ) 2 u 6 ( u 3 ) 2

v 20 ( v 4 ) 5 v 20 ( v 4 ) 5

m 12 ( m 8 ) 3 m 12 ( m 8 ) 3

n 8 ( n 6 ) 4 n 8 ( n 6 ) 4

( p 9 p 3 ) 5 ( p 9 p 3 ) 5

( q 8 q 2 ) 3 ( q 8 q 2 ) 3

( r 2 r 6 ) 3 ( r 2 r 6 ) 3

( m 4 m 7 ) 4 ( m 4 m 7 ) 4

( p r 11 ) 2 ( p r 11 ) 2

( a b 6 ) 3 ( a b 6 ) 3

( w 5 x 3 ) 8 ( w 5 x 3 ) 8

( y 4 z 10 ) 5 ( y 4 z 10 ) 5

( 2 j 3 3 k ) 4 ( 2 j 3 3 k ) 4

( 3 m 5 5 n ) 3 ( 3 m 5 5 n ) 3

( 3 c 2 4 d 6 ) 3 ( 3 c 2 4 d 6 ) 3

( 5 u 7 2 v 3 ) 4 ( 5 u 7 2 v 3 ) 4

( k 2 k 8 k 3 ) 2 ( k 2 k 8 k 3 ) 2

( j 2 j 5 j 4 ) 3 ( j 2 j 5 j 4 ) 3

( t 2 ) 5 ( t 4 ) 2 ( t 3 ) 7 ( t 2 ) 5 ( t 4 ) 2 ( t 3 ) 7

( q 3 ) 6 ( q 2 ) 3 ( q 4 ) 8 ( q 3 ) 6 ( q 2 ) 3 ( q 4 ) 8

( −2 p 2 ) 4 ( 3 p 4 ) 2 ( −6 p 3 ) 2 ( −2 p 2 ) 4 ( 3 p 4 ) 2 ( −6 p 3 ) 2

( −2 k 3 ) 2 ( 6 k 2 ) 4 ( 9 k 4 ) 2 ( −2 k 3 ) 2 ( 6 k 2 ) 4 ( 9 k 4 ) 2

( −4 m 3 ) 2 ( 5 m 4 ) 3 ( −10 m 6 ) 3 ( −4 m 3 ) 2 ( 5 m 4 ) 3 ( −10 m 6 ) 3

( −10 n 2 ) 3 ( 4 n 5 ) 2 ( 2 n 8 ) 2 ( −10 n 2 ) 3 ( 4 n 5 ) 2 ( 2 n 8 ) 2

In the following exercises, divide the monomials.

56 b 8 ÷ 7 b 2 56 b 8 ÷ 7 b 2

63 v 10 ÷ 9 v 2 63 v 10 ÷ 9 v 2

−88 y 15 ÷ 8 y 3 −88 y 15 ÷ 8 y 3

−72 u 12 ÷ 1 2 u 4 −72 u 12 ÷ 1 2 u 4

45 a 6 b 8 −15 a 10 b 2 45 a 6 b 8 −15 a 10 b 2

54 x 9 y 3 −18 x 6 y 15 54 x 9 y 3 −18 x 6 y 15

15 r 4 s 9 18 r 9 s 2 15 r 4 s 9 18 r 9 s 2

20 m 8 n 4 30 m 5 n 9 20 m 8 n 4 30 m 5 n 9

18 a 4 b 8 −27 a 9 b 5 18 a 4 b 8 −27 a 9 b 5

45 x 5 y 9 −60 x 8 y 6 45 x 5 y 9 −60 x 8 y 6

64 q 11 r 9 s 3 48 q 6 r 8 s 5 64 q 11 r 9 s 3 48 q 6 r 8 s 5

65 a 10 b 8 c 5 42 a 7 b 6 c 8 65 a 10 b 8 c 5 42 a 7 b 6 c 8

( 10 m 5 n 4 ) ( 5 m 3 n 6 ) 25 m 7 n 5 ( 10 m 5 n 4 ) ( 5 m 3 n 6 ) 25 m 7 n 5

( −18 p 4 q 7 ) ( −6 p 3 q 8 ) −36 p 12 q 10 ( −18 p 4 q 7 ) ( −6 p 3 q 8 ) −36 p 12 q 10

( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 2 b ) ( a 3 b ) ( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 2 b ) ( a 3 b )

( 4 u 2 v 5 ) ( 15 u 3 v ) ( 12 u 3 v ) ( u 4 v ) ( 4 u 2 v 5 ) ( 15 u 3 v ) ( 12 u 3 v ) ( u 4 v )

Mixed Practice

ⓐ 24 a 5 + 2 a 5 24 a 5 + 2 a 5 ⓑ 24 a 5 − 2 a 5 24 a 5 − 2 a 5 ⓒ 24 a 5 · 2 a 5 24 a 5 · 2 a 5 ⓓ 24 a 5 ÷ 2 a 5 24 a 5 ÷ 2 a 5

ⓐ 15 n 10 + 3 n 10 15 n 10 + 3 n 10 ⓑ 15 n 10 − 3 n 10 15 n 10 − 3 n 10 ⓒ 15 n 10 · 3 n 10 15 n 10 · 3 n 10 ⓓ 15 n 10 ÷ 3 n 10 15 n 10 ÷ 3 n 10

ⓐ p 4 · p 6 p 4 · p 6 ⓑ ( p 4 ) 6 ( p 4 ) 6

ⓐ q 5 · q 3 q 5 · q 3 ⓑ ( q 5 ) 3 ( q 5 ) 3

ⓐ y 3 y y 3 y ⓑ y y 3 y y 3

ⓐ z 6 z 5 z 6 z 5 ⓑ z 5 z 6 z 5 z 6

( 8 x 5 ) ( 9 x ) ÷ 6 x 3 ( 8 x 5 ) ( 9 x ) ÷ 6 x 3

( 4 y ) ( 12 y 7 ) ÷ 8 y 2 ( 4 y ) ( 12 y 7 ) ÷ 8 y 2

27 a 7 3 a 3 + 54 a 9 9 a 5 27 a 7 3 a 3 + 54 a 9 9 a 5

32 c 11 4 c 5 + 42 c 9 6 c 3 32 c 11 4 c 5 + 42 c 9 6 c 3

32 y 5 8 y 2 − 60 y 10 5 y 7 32 y 5 8 y 2 − 60 y 10 5 y 7

48 x 6 6 x 4 − 35 x 9 7 x 7 48 x 6 6 x 4 − 35 x 9 7 x 7

63 r 6 s 3 9 r 4 s 2 − 72 r 2 s 2 6 s 63 r 6 s 3 9 r 4 s 2 − 72 r 2 s 2 6 s

56 y 4 z 5 7 y 3 z 3 − 45 y 2 z 2 5 y 56 y 4 z 5 7 y 3 z 3 − 45 y 2 z 2 5 y

Everyday Math

Memory One megabyte is approximately 10 6 10 6 bytes. One gigabyte is approximately 10 9 10 9 bytes. How many megabytes are in one gigabyte?

Memory One gigabyte is approximately 10 9 10 9 bytes. One terabyte is approximately 10 12 10 12 bytes. How many gigabytes are in one terabyte?

Writing Exercises

Jennifer thinks the quotient a 24 a 6 a 24 a 6 simplifies to a 4 a 4 . What is wrong with her reasoning?

Maurice simplifies the quotient d 7 d d 7 d by writing d 7 d = 7 d 7 d = 7 . What is wrong with his reasoning?

When Drake simplified − 3 0 − 3 0 and ( −3 ) 0 ( −3 ) 0 he got the same answer. Explain how using the Order of Operations correctly gives different answers.

Robert thinks x 0 x 0 simplifies to 0. What would you say to convince Robert he is wrong?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Dividing Monomials

Learning outcomes.

• Divide monomials

Divide Monomials

In the previous module we reviewed all the properties of exponents. We will now use them to divide monomials. Later, you will use them to divide polynomials.

Find the quotient: [latex]56{x}^{5}\div 7{x}^{2}[/latex]

Find the quotient: [latex]63{x}^{8}\div 9{x}^{4}[/latex]

[latex]7{x}^{4}[/latex]  

Find the quotient: [latex]96{y}^{11}\div 6{y}^{8}[/latex]

[latex]16{y}^{3}[/latex]

When we divide monomials with more than one variable, we write one fraction for each variable.

Find the quotient: [latex]{\dfrac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}}[/latex]

Find the quotient: [latex]{\dfrac{-84{x}^{8}{y}^{3}}{7{x}^{10}{y}^{2}}}[/latex]

[latex]-{\dfrac{12y}{{x}^{2}}}[/latex]

Find the quotient: [latex]{\dfrac{-72{a}^{4}{b}^{5}}{-8{a}^{9}{b}^{5}}}[/latex]

[latex]{\dfrac{9}{{a}^{5}}}[/latex]

Find the quotient: [latex]{\dfrac{24{a}^{5}{b}^{3}}{48a{b}^{4}}}[/latex]

Find the quotient: [latex]{\dfrac{16{a}^{7}{b}^{6}}{24a{b}^{8}}}[/latex]

[latex]{\dfrac{2{a}^{6}}{3{b}^{2}}}[/latex]

Find the quotient: [latex]{\dfrac{27{p}^{4}{q}^{7}}{-45{p}^{12}{q}^{}}}[/latex]

[latex]-{\dfrac{3{q}^{6}}{5{p}^{8}}}[/latex]

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Find the quotient: [latex]{\dfrac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}}[/latex]

Be very careful to simplify [latex]{\dfrac{14}{21}}[/latex] by dividing out a common factor, and to simplify the variables by subtracting their exponents.

Find the quotient: [latex]{\dfrac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}}[/latex]

[latex]{\dfrac{4{y}^{2}}{7{x}^{4}}}[/latex]

Find the quotient: [latex]{\dfrac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}}[/latex]

[latex]{\dfrac{5}{8{m}^{5}{n}^{3}}}[/latex]

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction.

Find the quotient: [latex]{\dfrac{(3{x}^{3}{y}^{2})(10{x}^{2}{y}^{3})}{6{x}^{4}{y}^{5}}}[/latex]

Solution Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

Find the quotient: [latex]{\Large\frac{\left(3{x}^{4}{y}^{5}\right)\left(8{x}^{2}{y}^{5}\right)}{12{x}^{5}{y}^{8}}}[/latex]

[latex]2{xy}^{2}[/latex]

Find the quotient: [latex]{\Large\frac{\left(-6{a}^{6}{b}^{9}\right)\left(-8{a}^{5}{b}^{8}\right)}{-12{a}^{10}{b}^{12}}}[/latex]

[latex]-4{ab}^{5}[/latex]

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• Ex 1: Simplify Fractions. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/_2Wk7jXf3Ok . License : CC BY: Attribution
• Ex: Simplify Exponential Expressions Using the Power Property of Exponents. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/Hgu9HKDHTUA . License : CC BY: Attribution
• Question ID: 146014, 146148. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL
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RWM102: Algebra

Unit 7: operations with monomials.

As we have seen, algebra involves the use of variables to represent unknown quantities in equations. Here we will begin the study of expressions that primarily consist of variables.

The rules that govern operations with these expressions arise from the properties of operations with numbers, such as the distributive property and the order of operations. In this unit we focus on monomials, which are expressions that contain only one term. We will learn how to multiply, divide, and apply rules of exponents to monomials.

Completing this unit should take you approximately 3 hours.

Upon successful completion of this unit, you will be able to:

• apply the rules of exponents to simplify algebraic exponential expressions; and
• multiply, divide, and simplify the powers of monomials.

7.1: Algebraic Exponential Expressions

The first mathematical functions we will explore are the exponential functions of monomials.

Read this article through the section on the product of powers property. This page reviews writing and solving exponential expressions. Pay attention to how we multiply monomials with exponents, the products of powers property, and example 2.

After you read, complete questions 2, 3, 6, 7, 8, 25, 26, 27 and 28 in the practice set and check your answers.

7.2: Manipulating Exponents

In this section, we will look at important rules for manipulating exponents.

Read this article, which shows the rules for multiplying exponents, taking the power of an exponent, and taking the exponent of a product. It may help to write a list of these properties to keep track of them. Watch the video to see a few examples.

After you read, complete questions 2, 3, 6, 7, 14, 15, 25, 26, 33, and 34 in the practice set and check your answers.

7.3: Quotient of Exponents and Power of a Quotient

The next important operation is the quotient of exponents, which is what we use when dividing monomials with exponents.

Read this article and pay attention to the quotient of powers property and the power of a quotient property in examples 1 and 2.

After you read, complete questions 1, 2, 5, 6, 12, and 13 in the practice set and check your answers.

7.4: Negative Exponents

In this section, we learn how to use negative exponents and how to translate between negative exponents and fractions.

Watch these videos, which walk you through the logic of why negative exponents can be written as fractions.

After you watch, complete this assessment and check your answers.

7.5: Multiplying Monomials

Now that you are comfortable with the rules of exponents, you are ready to apply them to functions of monomials. In this section, we will discuss multiplying monomials.

Read the section on multiplying monomials and the solution for example 6.26.

After you read, complete example 6.27 and check your answer.

7.6: Dividing Monomials

The last function we must apply to monomials is division.

This article gives an excellent review of the properties of monomials and exponents, as well as how to divide monomials. Review the summary of exponent properties, which goes over everything we discussed in this unit. After you read, complete examples 6.72 through 6.75 and check your work.

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4.7: Dividing Polynomials

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Learning Objective

• Divide by a monomial.

Dividing by a Monomial

Recall the quotient rule for exponents: if \(x\) is nonzero and \(m\) and \(n\) are positive integers, then

\[\frac{x^{m}}{x^{n}}=x^{m-n}\]

In other words, when dividing two expressions with the same base, subtract the exponents. This rule applies when dividing a monomial by a monomial. In this section, we will assume that all variables in the denominator are nonzero.

Example \(\PageIndex{1}\)

\(\frac{28y^{3}}{7y}\).

Divide the coefficients and subtract the exponents of the variable \(y\).

\(\begin{aligned} \frac{28y^{3}}{7y}&=\frac{28}{7}y^{3-1} \\ &=4y^{2} \end{aligned}\)

Example \(\PageIndex{2}\)

\(\frac{24x^{7}y^{5}}{8x^{3}y^{2}}\).

Divide the coefficients and apply the quotient rule by subtracting the exponents of the like bases.

\(\begin{aligned} \frac{24x^{7}y^{5}}{8x^{3}y^{2}}&=\frac{24}{8}x^{7-3}y^{5-2} \\ &=3x^{4}y^{3} \end{aligned}\)

\(3x^{4}y^{3}\)

When dividing a polynomial by a monomial, we may treat the monomial as a common denominator and break up the fraction using the following property:

\[\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\]

Applying this property results in terms that can be treated as quotients of monomials.

Example \(\PageIndex{3}\)

\(\frac{−5x^{4}+25x^{3}−15x^{2}}{5x^{2}}\).

Break up the fraction by dividing each term in the numerator by the monomial in the denominator and then simplify each term.

\(-x^{2}+5x-3\cdot 1\)

Check your division by multiplying the answer, the quotient, by the monomial in the denominator, the divisor, to see if you obtain the original numerator, the dividend.

\[\color{Cerulean}{\frac{dividend}{divisor}=quotient}\]

\[\color{Cerulean}{dividend=divisor\cdot quotient}\]

\(\begin{aligned} 5x^{2}\cdot (-x^{2}+5x-3) &=\color{Cerulean}{5x^{2}}\color{black}{\cdot (-x^{2})+}\color{Cerulean}{5x^{2}}\color{black}{\cdot 5x-}\color{Cerulean}{5x^{2}}\color{black}{\cdot 3} \\ &=-5x^{4}+25x^{3}-15x^{2}\quad\color{Cerulean}{\checkmark} \end{aligned}\)

Example \(\PageIndex{4}\)

\(\frac{9a^{4}b−7a^{3}b^{2}+3a^{2}b}{−3a^{2}b}\).

\(-3a^{2}+\frac{7}{3}ab-1\). The check is optional and is left to the reader.

Exercise \(\PageIndex{1}\)

\((16x^{5}−8x^{4}+5x^{3}+2x^{2})÷(2x^{2})\).

\(8x^{3}−4x^{2}+\frac{5}{2}x+1\)

Key Takeaways

• When dividing by a monomial, divide all terms in the numerator by the monomial and then simplify each term. To simplify each term, divide the coefficients and apply the quotient rule for exponents.

Exercise \(\PageIndex{3}\) Dividing by a Monomial

• \(\frac{81y^{5}}{9y^{2}}\)
• \(\frac{36y^{9}}{9y^{3}}\)
• \(\frac{52x^{2}y}{4xy}\)
• \(\frac{24xy^{5}}{2xy^{4}}\)
• \(\frac{25x^{2}y^{5}z^{3}}{5xyz}\)
• \(\frac{−77x^{4}y^{9}z^{2}}{2x^{3}y^{3}z}\)
• \(\frac{125a^{3}b^{2}c}{−10abc}\)
• \(\frac{36a^{2}b^{3}c^{5}}{−6a^{2}b^{2}c^{3}}\)
• \(\frac{9x^{2}+27x−3}{3}\)
• \(\frac{10x^{3}−5x^{2}+40x−15}{5}\)
• \(\frac{20x^{3}−10x^{2}+30x}{2x}\)
• \(\frac{10x^{4}+8x^{2}−6x}{24x}\)
• \(\frac{−6x^{5}−9x^{3}+3x}{−3x}\)
• \(\frac{36a^{12}−6a^{9}+12a^{5}}{−12a^{5}}\)
• \(\frac{−12x^{5}+18x^{3}−6x^{2}}{−6x^{2}}\)
• \(\frac{−49a^{8}+7a^{5}−21a^{3}}{7a^{3}}\)
• \(\frac{9x^{7}−6x^{4}+12x^{3}−x^{2}}{3x^{2}}\)
• \(\frac{8x^{9}+16x^{7}−24x^{4}+8x^{3}}{−8x^{3}}\)
• \(\frac{16a^{7}−32a^{6}+20a^{5}−a^{4}}{4a^{4}}\)
• \(\frac{5a^{6}+2a^{5}+6a^{3}−12a^{2}}{3a^{2}}\)
• \(\frac{−4x^{2}y^{3}+16x^{7}y^{8}−8x^{2}y^{5}}{−4x^{2}y^{3}}\)
• \(\frac{100a^{10}b^{30}c^{5}−50a^{20}b^{5}c^{40}+20a^{5}b^{20}c^{10}}{10a^{5}b^{5}c^{5}}\)
• Find the quotient of \(−36x^{9}y^{7}\) and \(2x^{8}y^{5}\).
• Find the quotient of \(144x^{3}y^{10}z^{2}\) and \(−12x^{3}y^{5}z\).
• Find the quotient of \(3a^{4}−18a^{3}+27a^{2}\) and \(3a^{2}\).
• Find the quotient of \(64a^{2}bc^{3}−16a^{5}bc^{7}\) and \(4a^{2}bc^{3}\).

1. \(9y^{3}\)

5. \(5xy^{4}z^{2}\)

7. \(−\frac{25}{2}a^{2}b\)

9. \(3x^{2}+9x−1\)

11. \(10x^{2}−5x+15\)

13. \(2x^{4}+3x^{2}−1\)

15. \(2x^{3}−3x+1\)

17. \(3x^{5}−2x^{2}+4x−\frac{1}{3}\)

19. \(4a^{3}−8a^{2}+5a−\frac{1}{4}\)

21. \(−4x^{5}y^{5}+2y^{2}+1\)

23. \(−18xy^{2}\)

25. \(a^{2}−6a+9\)