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( 8 x ) 3 means ( 8 x ) ( 8 x ) ( 8 x ) since the parentheses indicate that the exponent 3 is directly connected to the factor 8 x . Remember that the grouping symbols (   ) indicate that the quantities inside are to be considered as one single number.

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34 ( a + 1 ) 2 means 34 ( a + 1 ) ( a + 1 ) since the exponent 2 applies only to the factor ( a + 1 ) .

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Practice set b

Write each of the following without exponents.

( 4 a ) 3

( 4 a ) ( 4 a ) ( 4 a )

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Sample set c

Select a number to show that ( 2 x ) 2 is not always equal to 2 x 2 .

Suppose we choose x to be 5. Consider both ( 2 x ) 2 and 2 x 2 .

( 2 x ) 2 2 x 2 ( 2 5 ) 2 2 5 2 ( 10 ) 2 2 25 100 50

Notice that ( 2 x ) 2 = 2 x 2 only when x = 0 .

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Practice set c

Select a number to show that ( 5 x ) 2 is not always equal to 5 x 2 .

Select x = 3 . Then ( 5 3 ) 2 = ( 15 ) 2 = 225 , but 5 3 2 = 5 9 = 45 .     225 45 .

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Reading exponential notation

In x n ,

Base

x is the base

Exponent

n is the exponent

Power

The number represented by x n is called a power .

x To the n Th power

The term x n is read as " x to the n th power," or more simply as " x to the n th."

x Squared and x Cubed

The symbol x 2 is often read as " x squared," and x 3 is often read as " x cubed." A natural question is "Why are geometric terms appearing in the exponent expression?" The answer for x 3 is this: x 3 means x x x . In geometry, the volume of a rectangular box is found by multiplying the length by the width by the depth. A cube has the same length on each side. If we represent this length by the letter x then the volume of the cube is x x x , which, of course, is described by x 3 . (Can you think of why x 2 is read as x squared?)

Cube with
length = x
width = x
depth = x
Volume = x x x = x 3

A cube with length of side equal to x.

The order of operations

In Section [link] we were introduced to the order of operations. It was noted that we would insert another operation before multiplication and division. We can do that now.

    The order of operations

  1. Perform all operations inside grouping symbols beginning with the innermost set.
  2. Perform all exponential operations as you come to them, moving left-to-right.
  3. Perform all multiplications and divisions as you come to them, moving left-to-right.
  4. Perform all additions and subtractions as you come to them, moving left-to-right.

Sample set d

Use the order of operations to simplify each of the following.

5 2 + 3 2 + 10 = 25 + 9 + 10 = 44

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2 2 + ( 5 ) ( 8 ) 1 = 4 + ( 5 ) ( 8 ) 1 = 4 + 40 1 = 43

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7 6 4 2 + 1 5 = 7 6 16 + 1 = 42 16 + 1 = 27

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( 2 + 3 ) 3 + 7 2 3 ( 4 + 1 ) 2 = ( 5 ) 3 + 7 2 3 ( 5 ) 2 = 125 + 49 3 ( 25 ) = 125 + 49 75 = 99

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[ 4 ( 6 + 2 ) 3 ] 2 = [ 4 ( 8 ) 3 ] 2 = [ 4 ( 512 ) ] 2 = [ 2048 ] 2 = 4 , 194 , 304

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6 ( 3 2 + 2 2 ) + 4 2 = 6 ( 9 + 4 ) + 4 2 = 6 ( 13 ) + 4 2 = 6 ( 13 ) + 16 = 78 + 16 = 94

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6 2 + 2 2 4 2 + 6 2 2 + 1 3 + 8 2 10 2 ( 19 ) ( 5 ) = 36 + 4 16 + 6 4 + 1 + 64 100 95 = 36 + 4 16 + 24 + 1 + 64 100 95 = 40 40 + 65 5 = 1 + 13 = 14

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Practice set d

Use the order of operations to simplify the following.

1 4 + ( 2 2 + 4 ) 2 ÷ 2 3

9

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[ 6 ( 10 2 3 ) ] 2 10 2 6 2

8

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5 2 + 6 2 10 1 + 4 2 + 0 4 0 5 7 2 6 2 3

3

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Exercises

For the following problems, write each of the quantities using exponential notation.

3 squared times y to the fifth

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a cubed minus ( b + 7 ) squared

a 3 ( b + 7 ) 2

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( 21 x ) cubed plus ( x + 5 ) to the seventh

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2 3 3 3 3 x x y y y y y

2 ( 3 4 ) x 2 y 5

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2 2 5 6 6 6 x y y z z z w w w w

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7 x x ( a + 8 ) ( a + 8 )

7 x 2 ( a + 8 ) 2

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10 x y y ( c + 5 ) ( c + 5 ) ( c + 5 )

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4 x 4 x 4 x 4 x 4 x

( 4 x ) 5 or 4 5 x 5

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( 9 a ) ( 9 a ) ( 9 a ) ( 9 a )

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( 7 ) ( 7 ) ( 7 ) a a b b b a ( 7 ) b a a b

( 7 ) 4 a 5 b 5

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( a 10 ) ( a 10 ) ( a + 10 )

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( z + w ) ( z + w ) ( z + w ) ( z w ) ( z w )

( z + w ) 3 ( z w ) 2

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3 x y x x y ( x + 1 ) ( x + 1 ) ( x + 1 )

3 x 3 y 2 ( x + 1 ) 3

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For the following problems, expand the quantities so that no exponents appear.

8 x 3 y 2

8 · x · x · x · y · y

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( 9 a 3 b 2 ) 3

( 9 a a a b b ) ( 9 a a a b b ) ( 9 a a a b b ) or 9 · 9 · 9 a a a a a a a a a b b b b b b

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10 a 3 b 2 ( 3 c ) 2

10 a a a b b ( 3 c ) ( 3 c )  or 10 · 3 · 3 a a a b b c c

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( a + 10 ) 2 ( a 2 + 10 ) 2

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( x 2 y 2 ) ( x 2 + y 2 )

( x x y y ) ( x x + y y )

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For the following problems, select a number (or numbers) to show that

( 5 x ) 2 is not generally equal to 5 x 2 .

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( 7 x ) 2 is not generally equal to 7 x 2 .

Select x = 2. Then, 196 28.

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( a + b ) 2 is not generally equal to a 2 + b 2 .

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For what real number is ( 6 a ) 2 equal to 6 a 2 ?

zero

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For what real numbers, a and b , is ( a + b ) 2 equal to a 2 + b 2 ?

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Use the order of operations to simplify the quantities for the following problems.

( 4 + 3 ) 2 + 1 ÷ ( 2 5 )

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( 2 4 + 2 5 2 3 5 ) 2 ÷ 4 2

4

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1 6 + 0 8 + 5 2 ( 2 + 8 ) 3

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( 7 ) ( 16 ) 9 2 + 4 ( 1 1 + 3 2 )

71

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( 1 + 6 ) 2 + 2 19

51 19

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6 2 1 5 + 4 3 + ( 2 ) ( 3 ) 10

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5 [ 8 2 9 ( 6 ) ] 2 5 7 + 7 2 4 2 2 4 5

5

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( 2 + 1 ) 3 + 2 3 + 1 3 6 2 15 2 [ 2 ( 5 ) ] 2 5 5 2

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6 3 2 10 2 2 2 + 18 ( 2 3 + 7 2 ) 2 ( 19 ) 3 3

1070 11 or 97. 27 ¯

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Exercises for review

( [link] ) Use algebraic notation to write the statement "a number divided by eight, plus five, is equal to ten."

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( [link] ) Draw a number line that extends from 5 to 5 and place points at all real numbers that are strictly greater than 3 but less than or equal to 2.

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a closed circle at two, and an open circle at negative three. These circles are connected by a black line.

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( [link] ) Is every integer a whole number?

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( [link] ) Use the commutative property of multiplication to write a number equal to the number y x .

x y

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( [link] ) Use the distributive property to expand 3 ( x + 6 ) .

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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?
Aislinn Reply
cm
tijani
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Siyaka Reply
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
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
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
Krampah Reply
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.
Sahid Reply
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
Samuel Reply
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?
Joseph Reply
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
Ryan
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Maurice
answer
Magreth
progressive wave
Magreth
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hi
Mujahid
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?
yasuo Reply
Who can show me the full solution in this problem?
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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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