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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses exponents and roots. By the end of the module students should be able to understand and be able to read exponential notation, understand the concept of root and be able to read root notation, and use a calculator having the y x key to determine a root.

Section overview

  • Exponential Notation
  • Reading Exponential Notation
  • Roots
  • Reading Root Notation
  • Calculators

Exponential notation

Exponential notation

We have noted that multiplication is a description of repeated addition. Exponen­tial notation is a description of repeated multiplication.

Suppose we have the repeated multiplication

8 8 8 8 8 size 12{8 cdot 8 cdot 8 cdot 8 cdot 8} {}

Exponent

The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, 8 5 , in this case. The superscript is called an exponent .

The function of an exponent

An exponent records the number of identical factors that are repeated in a multiplication.

Sample set a

Write the following multiplication using exponents.

3 3 size 12{3 cdot 3} {} . Since the factor 3 appears 2 times, we record this as

3 2 size 12{3 rSup { size 8{2} } } {}

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62 62 62 62 62 62 62 62 62 size 12{"62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62"} {} . Since the factor 62 appears 9 times, we record this as

62 9 size 12{"62" rSup { size 8{9} } } {}

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Expand (write without exponents) each number.

12 4 size 12{"12" rSup { size 8{4} } } {} . The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,

12 4 = 12 12 12 12 size 12{"12" rSup { size 8{4} } ="12" cdot "12" cdot "12" cdot "12"} {}

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706 3 size 12{"706" rSup { size 8{3} } } {} . The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,

706 3 = 706 706 706 size 12{"706" rSup { size 8{3} } ="706" cdot "706" cdot "706"} {}

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Practice set a

Write the following using exponents.

37 37 size 12{"37" cdot "37"} {}

37 2 size 12{"37" rSup { size 8{2} } } {}

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16 16 16 16 16 size 12{"16" cdot "16" cdot "16" cdot "16" cdot "16"} {}

16 5 size 12{"16" rSup { size 8{5} } } {}

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9 9 9 9 9 9 9 9 9 9 size 12{9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9} {}

9 10 size 12{9 rSup { size 8{"10"} } } {}

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Write each number without exponents.

85 3 size 12{"85" rSup { size 8{3} } } {}

85 85 85 size 12{"85" cdot "85" cdot "85"} {}

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4 7 size 12{4 rSup { size 8{7} } } {}

4 4 4 4 4 4 4 size 12{4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4} {}

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1, 739 2 size 12{1,"739" rSup { size 8{2} } } {}

1, 739 1, 739 size 12{1,"739" cdot 1,"739"} {}

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Reading exponential notation

In a number such as 8 5 size 12{8 rSup { size 8{5} } } {} ,

Base

8 is called the base .

Exponent, power

5 is called the exponent , or power . 8 5 size 12{8 rSup { size 8{5} } } {} is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."

Squared

When a whole number is raised to the second power, it is said to be squared . The number 5 2 size 12{5 rSup { size 8{2} } } {} can be read as

5 to the second power, or
5 to the second, or
5 squared.

Cubed

When a whole number is raised to the third power, it is said to be cubed . The number 5 3 size 12{5 rSup { size 8{3} } } {} can be read as

5 to the third power, or
5 to the third, or
5 cubed.

When a whole number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number 5 8 size 12{5 rSup { size 8{8} } } {} can be read as

5 to the eighth power, or just
5 to the eighth.

Roots

In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication.

Square root

We know that 49 = 7 2 size 12{"49"=7 rSup { size 8{2} } } {} , that is, 49 = 7 7 size 12{"49"=7 cdot 7} {} . Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49.

Cube root

We know that 8 = 2 3 size 12{8=2 rSup { size 8{3} } } {} , that is, 8 = 2 2 2 size 12{8=2 cdot 2 cdot 2} {} . Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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