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By the end of this section, you will be able to:
  • Use the definition of a negative exponent
  • Simplify expressions with integer exponents
  • Convert from decimal notation to scientific notation
  • Convert scientific notation to decimal form
  • Multiply and divide using scientific notation

Before you get started, take this readiness quiz.

  1. What is the place value of the 6 in the number 64,891 ?
    If you missed this problem, review [link] .
  2. Name the decimal: 0.0012 .
    If you missed this problem, review [link] .
  3. Subtract: 5 ( −3 ) .
    If you missed this problem, review [link] .

Use the definition of a negative exponent

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

Quotient property for exponents

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

a m a n = a m n , m > n and a m a n = 1 a n m , n > m

What if we just subtract exponents regardless of which is larger?

Let’s consider x 2 x 5 .

We subtract the exponent in the denominator from the exponent in the numerator.

x 2 x 5 x 2 5 x −3

We can also simplify x 2 x 5 by dividing out common factors:

Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.

This implies that x −3 = 1 x 3 and it leads us to the definition of a negative exponent .

Negative exponent

If n is an integer and a 0 , then a n = 1 a n .

The negative exponent    tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent    and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression x −3 , we will take one more step and write 1 x 3 . The answer is considered to be in simplest form when it has only positive exponents.

Simplify: 4 −2 10 −3 .

Solution


  1. 4 −2 Use the definition of a negative exponent, a n = 1 a n . 1 4 2 Simplify. 1 16


  2. 10 −3 Use the definition of a negative exponent, a n = 1 a n . 1 10 3 Simplify. 1 1000
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Simplify: 2 −3 10 −7 .

1 8 1 10 7

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Simplify: 3 −2 10 −4 .

1 9 1 10,000

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In [link] we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

1 a n Use the definition of a negative exponent, a n = 1 a n . 1 1 a n Simplify the complex fraction. 1 · a n 1 Multiply. a n

This leads to the Property of Negative Exponents.

Property of negative exponents

If n is an integer and a 0 , then 1 a n = a n .

Simplify: 1 y −4 1 3 −2 .

Solution


  1. 1 y −4 Use the property of a negative exponent, 1 a n = a n . y 4


  2. 1 3 −2 Use the property of a negative exponent, 1 a n = a n . 3 2 Simplify. 9
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Simplify: 1 p −8 1 4 −3 .

p 8 64

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Simplify: 1 q −7 1 2 −4 .

q 7 16

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Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

( 3 4 ) −2 Use the definition of a negative exponent, a n = 1 a n . 1 ( 3 4 ) 2 Simplify the denominator. 1 9 16 Simplify the complex fraction. 16 9 But we know that 16 9 is ( 4 3 ) 2 . This tells us that: ( 3 4 ) −2 = ( 4 3 ) 2

Questions & Answers

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Practice Key Terms 2

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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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