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( 9 y ) 4 = 9 ( y 4 ) Both represent the same product .

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Practice set b

Fill in the ( ) to make each statement true. Use the associative properties.

( 9 + 2 ) + 5 = 9 + ( )

2 + 5

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x + ( 5 + y ) = ( ) + y

x + 5

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( 11 a ) 6 = 11 ( )

a 6

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[ ( 7 m 2 ) ( m + 3 ) ] ( m + 4 ) = ( 7 m 2 ) [ ( ) ( ) ]

( m + 3 ) ( m + 4 )

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Sample set c

Simplify (rearrange into a simpler form): 5 x 6 b 8 a c 4 .

According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.

5 6 8 4 x b a c 960 x b a c Multiply the numbers . 960 a b c x By convention, we will, when possible, write all letters in alphabetical order .

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Practice set c

Simplify each of the following quantities.

6 b 8 a c z 4 5

960 a b c z

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4 p 6 q r 3 ( a + b )

72 p q r ( a + b )

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The distributive properties

When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.

4 + 4 + 4 = 3 4

Notice that there are three 4’s, that is, 4 appears 3 times . Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.

When a number a is added repeatedly n times, we have
a + a + a + + a a appears n times
Then, using multiplication as a description for repeated addition, we can replace
a + a + a + + a n times with n a

For example:

x + x + x + x can be written as 4 x since x is repeatedly added 4 times.

x + x + x + x = 4 x

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r + r can be written as 2 r since r is repeatedly added 2 times.

r + r = 2 r

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The distributive property involves both multiplication and addition. Let’s rewrite 4 ( a + b ) . We proceed by reading 4 ( a + b ) as a multiplication: 4 times the quantity ( a + b ) . This directs us to write

4 ( a + b ) = ( a + b ) + ( a + b ) + ( a + b ) + ( a + b ) = a + b + a + b + a + b + a + b

Now we use the commutative property of addition to collect all the a ' s together and all the b ' s together.

4 ( a + b ) = a + a + a + a 4 a ' s + b + b + b + b 4 b ' s

Now, using multiplication as a description for repeated addition, we have

4 ( a + b ) = 4 a + 4 b

We have distributed the 4 over the sum to both a and b .

The product of four and the expression, a plus b, is equal to four a plus four b. The distributive property is shown by the arrows from four to each term of expression a plus b in the product.

The distributive property

a ( b + c ) = a b + a c ( b + c ) a = a b + a c

The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.

Sample set d

Use the distributive property to rewrite each of the following quantities.

Practice set d

What property of real numbers justifies
a ( b + c ) = ( b + c ) a ?

the commutative property of multiplication

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Use the distributive property to rewrite each of the following quantities.

The identity properties

Additive identity

The number 0 is called the additive identity since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, 6 + 0 = 6 .

Multiplicative identity

The number 1 is called the multiplicative identity since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example 6 1 = 6 .

We summarize the identity properties as follows.

ADDITIVE IDENTITY PROPERTY MULTIPLICATIVE IDENTITY PROPERTY If a is a real number, then If a is a real number, then a + 0 = a and 0 + a = a a 1 = a and 1 a = a

The inverse properties

Additive inverses

When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to 3 the result is 0, that is, 3 + ( 3 ) = 0 . The numbers 3 and 3 are additive inverses of each other.

Multiplicative inverses

When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when 6 and 1 6 are multiplied together, the result is 1, that is, 6 1 6 = 1 . The numbers 6 and 1 6 are multiplicative inverses of each other.

We summarize the inverse properties as follows.

    The inverse properties

  1. If a is any real number, then there is a unique real number a , such that
    a + ( a ) = 0 and a + a = 0
    The numbers a and a are called additive inverses of each other.
  2. If a is any nonzero real number, then there is a unique real number 1 a such that
    a 1 a = 1 and 1 a a = 1
    The numbers a and 1 a are called multiplicative inverses of each other.

Expanding quantities

When we perform operations such as 6 ( a + 3 ) = 6 a + 18 , we say we are expanding the quantity 6 ( a + 3 ) .

Exercises

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.

( x + 16 ) ( a + 7 )

( a + 7 ) ( x + 16 )

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5 ( 6 h + 1 )

( 6 h + 1 ) 5

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k ( 10 a b )

( 10 a b ) k

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( 16 ) ( 4 )

( 4 ) ( 16 )

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Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.

1 u 3 r 2 z 5 m 1 n

30 m n r u z

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6 d 4 e 1 f 2 ( g + 2 h )

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( 1 2 ) d ( 1 4 ) e ( 1 2 ) a

1 16 a d e

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3 ( a + 6 ) 2 ( a 9 ) 6 b

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1 ( x + 2 y ) ( 6 + z ) 9 ( 3 x + 5 y )

9 ( x + 2 y ) ( 6 + z ) ( 3 x + 5 y )

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For the following problems, use the distributive property to expand the quantities.

z ( x + 9 w )

x z + 9 w z

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( 8 + 2 f ) g

8 g + 2 f g

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15 x ( 2 y + 3 z )

30 x y + 45 x z

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z ( x + y + m )

x z + y z + m z

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( x + 10 ) ( a + b + c )

a x + b x + c x + 10 a + 10 b + 10 c

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Use a calculator. 21.5 ( 16.2 a + 3.8 b + 0.7 c )

348.3 a + 81.7 b + 15.05 c

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2 z t ( L m + 8 k )

2 L m z t + 16 k z t

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Exercises for review

( [link] ) Find the value of 4 2 + 5 ( 2 4 6 ÷ 3 ) 2 5 .

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( [link] ) Is the statement 3 ( 5 3 3 5 ) + 6 2 3 4 < 0 true or false?

false

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( [link] ) Draw a number line that extends from 2 to 2 and place points at all integers between and including 2 and 3.

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( [link] ) Replace the with the appropriate relation symbol ( < , > ) . 7 3 .

<

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( [link] ) What whole numbers can replace x so that the statement 2 x < 2 is true?

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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