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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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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