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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
7hours 36 min - 4hours 50 min
Tanis Reply

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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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