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We conclude that there is no answer to 4 ÷ 0 and so we say that division by 0 is undefined.

Division by zero

For any real number a , except 0, a 0 and a ÷ 0 are undefined.

Division by zero is undefined.

We summarize the properties of zero below.

Properties of zero

Multiplication by Zero: For any real number a ,

a · 0 = 0 0 · a = 0 The product of any number and 0 is 0.

Division of Zero, Division by Zero: For any real number a , a 0

0 a = 0 Zero divided by any real number, except itself is zero. a 0 is undefined Division by zero is undefined.

Simplify: −8 · 0 0 −2 −32 0 .

Solution


−8 · 0 The product of any real number and 0 is 0. 0


0 −2 Zero divided by any real number, except itself, is 0. 0


−32 0 Division by 0 is undefined. Undefined

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Simplify: −14 · 0 0 −6 −2 0 .

0 0 undefined

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Simplify: 0 ( −17 ) 0 −10 −5 0 .

0 0 undefined

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We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Simplify: 0 n + 5 , where n 5 10 3 p 0 , where 10 3 p 0 .

Solution


0 n + 5 Zero divided by any real number except itself is 0. 0


10 3 p 0 Division by 0 is undefined. Undefined

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Simplify: −84 n + ( −73 n ) + 84 n .

Solution

−84 n + ( −73 n ) + 84 n Notice that the first and third terms are opposites; use the commutative property of addition to re-order the terms. −84 n + 84 n + ( −73 n ) Add left to right. 0 + ( −73 ) Add. −73 n

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Simplify: −27 a + ( −48 a ) + 27 a .

−48 a

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Simplify: 39 x + ( −92 x ) + ( −39 x ) .

−92 x

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Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.

Simplify: 7 15 · 8 23 · 15 7 .

Solution

7 15 · 8 23 · 15 7 Notice the first and third terms are reciprocals, so use the commutative property of multiplication to re-order the factors. 7 15 · 15 7 · 8 23 Multiply left to right. 1 · 8 23 Multiply. 8 23

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Simplify: 9 16 · 5 49 · 16 9 .

5 49

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Simplify: 6 17 · 11 25 · 17 6 .

11 25

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Simplify: 0 m + 7 , where m 7 18 6 c 0 , where 18 6 c 0 .

0 undefined

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Simplify: 0 d 4 , where d 4 15 4 q 0 , where 15 4 q 0 .

0 undefined

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Simplify: 3 4 · 4 3 ( 6 x + 12 ) .

Solution

3 4 · 4 3 ( 6 x + 12 ) There is nothing to do in the parentheses, so multiply the two fractions first—notice, they are reciprocals. 1 ( 6 x + 12 ) Simplify by recognizing the multiplicative identity. 6 x + 12

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Simplify: 2 5 · 5 2 ( 20 y + 50 ) .

20 y + 50

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Simplify: 3 8 · 8 3 ( 12 z + 16 ) .

12 z + 16

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Simplify expressions using the distributive property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property .

Distributive property

If a , b , c are real numbers, then a ( b + c ) = a b + a c Also, ( b + c ) a = b a + c a a ( b c ) = a b a c ( b c ) a = b a c a

Back to our friends at the movies, we could find the total amount of money they need like this:

3 ( 9.25 ) 3 ( 9 + 0.25 ) 3 ( 9 ) + 3 ( 0.25 ) 27 + 0.75 27.75

In algebra, we use the distributive property to remove parentheses as we simplify expressions.

For example, if we are asked to simplify the expression 3 ( x + 4 ) , the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in [link] .

Questions & Answers

where's the answers?
Ed Reply
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
What does 3 times your weight right now
Cherokee Reply
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Kendra Reply
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
Quadratic equations involving factorization
Winner Reply
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Nana
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Nana
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Nana
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Joan
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Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
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Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Ok cool answer peggie
Frank
thanks
Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
Sandra Reply
What is c+4=8
Penny Reply
2
Letha
4
Lolita
4
Rich
4
thinking
C+4=8 -4 -4 C =4
thinking
I need to study
Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
Practice Key Terms 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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