What is an exponent?

Exponents are a
shorthand way to show
a larger number.
Where is an exponent
located at?
Exponents are sometimes
referred to as “powers.”

Base

6

4

Exponent
How do you read an
exponent?
A number with an exponent
is said to be "raised to the
power" of that exponent.
Let’s look at...
32 = three raised to the second
power or three squared
53 = five raised to the third
power or five cubed
74 = seven raised...
What does an
exponent mean?
We will use an example of
how the secret spread fast
to explain what does an
exponent mean.
How does a secret
spread so fast?

• The person with a secret tells a
friend.
• The friend promises never to tell
anyone.
...
• The two new friends decide to
tell two more friends.
• This pattern occurs over and
over until many people have
been tol...
Here is how a secret
can spread
Round 1

Round 2

Look how
many
people
now know
the
secret!
3 x 3 =
9

9 people
know!
I ha...
In an exponential expression, the base is the factor,
and the exponent tells the amount of times to
multiple that number b...
LAWS OF
EXPONENTS
• Recall: A number in exponential form
has a base and an exponent. The
exponent indicates how many times
the base is used ...
NOTES
The laws of exponents show how to
SIMPLIFY
expressions in exponential form.
NOTES

In the next few slides, a and b are real
numbers and m and n are integers.
Product of powers
• To multiply powers with the same
base, add the exponents.

a ∙a = a
m

n

m+n
Example: 2 × = 2
2
3

4

3+ 4

=2

7

Proof: 23 × 4 = ( 2 × × ) × 2 × × × ) =
2
2 2 ( 2 2 2
2 × × × × × × =2
2 2 2 2 2 2

...
Power of a power
• To raise a number in exponential form
to a power, multiply the exponents.

(a ) = a
m n

m∙n
Example: ( 4 ) = 4 = 4
2
3

3×2

6
Power of a product
• To find a power of a product, find the
power of each factor and multiply.

(ab) = a b
m

m

m
Example: 36 = 6 = ( 2 ×3) = 2 ×3
2

Proof: 2 ×3 = 4 ×9 = 36
2

2

2

2

2
Power of zero
• Any nonzero number raised to the
power of zero is one.

a =1
0
Quotient of powers
• To find the quotient of powers with
the same base, subtract the
exponents.

a =a
n
a
m

m-n
•For example…

3 ÷3 =3
2

2

2-2

3 = 3 • 3 =9 =1
2

3 3•3
2

9

=3

0
Power of a quotient
To raise a quotient to a power, raise
both the numerator and denominator
to that power.

(a/b) = a /b
...
3

3

2
2
Example:  ÷ = 3
7
7
Reciprocals
• To change a sign of an exponent, move
the expression to the denominator of
a fraction, or to the numerator.
...
1 1
Example #1: 2 = 3 =
2 8
−3

3

1 5 3
Example #2: − 3 = = 5 = 125
5
1
references
Maslijr, 2013, Laws of exponents, viewed 05 March 2014, from
http://www.slideshare.net/masljr/laws-of-exponents...
THE END
Basics about exponents
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Basics about exponents

  1. 1. What is an exponent? Exponents are a shorthand way to show a larger number.
  2. 2. Where is an exponent located at? Exponents are sometimes referred to as “powers.” Base 6 4 Exponent
  3. 3. How do you read an exponent? A number with an exponent is said to be "raised to the power" of that exponent. Let’s look at some examples. . . .
  4. 4. 32 = three raised to the second power or three squared 53 = five raised to the third power or five cubed 74 = seven raised to the fourth power
  5. 5. What does an exponent mean? We will use an example of how the secret spread fast to explain what does an exponent mean.
  6. 6. How does a secret spread so fast? • The person with a secret tells a friend. • The friend promises never to tell anyone. • That same friend breaks the promise and tells two more friends. • The two new friends tell two new friends.
  7. 7. • The two new friends decide to tell two more friends. • This pattern occurs over and over until many people have been told. • By the end of the day it is no longer a secret!
  8. 8. Here is how a secret can spread Round 1 Round 2 Look how many people now know the secret! 3 x 3 = 9 9 people know! I have a secret and I tell 3 of my best friends. My 3 friends each tell 3 more people.
  9. 9. In an exponential expression, the base is the factor, and the exponent tells the amount of times to multiple that number by itself. (That is a mouthful!) Your BASE = 3 3 4 Your EXPONENT = 4 • A base is a number that can be expressed using an exponent. • An exponent is the small number and is referred to as a “power.” 3 means 3 x 3 x 3 or 27 4
  10. 10. LAWS OF EXPONENTS
  11. 11. • Recall: A number in exponential form has a base and an exponent. The exponent indicates how many times the base is used as a factor. • In its exponential form: a is the BASE and b is the EXPONENT. ab
  12. 12. NOTES The laws of exponents show how to SIMPLIFY expressions in exponential form.
  13. 13. NOTES In the next few slides, a and b are real numbers and m and n are integers.
  14. 14. Product of powers • To multiply powers with the same base, add the exponents. a ∙a = a m n m+n
  15. 15. Example: 2 × = 2 2 3 4 3+ 4 =2 7 Proof: 23 × 4 = ( 2 × × ) × 2 × × × ) = 2 2 2 ( 2 2 2 2 × × × × × × =2 2 2 2 2 2 2 7
  16. 16. Power of a power • To raise a number in exponential form to a power, multiply the exponents. (a ) = a m n m∙n
  17. 17. Example: ( 4 ) = 4 = 4 2 3 3×2 6
  18. 18. Power of a product • To find a power of a product, find the power of each factor and multiply. (ab) = a b m m m
  19. 19. Example: 36 = 6 = ( 2 ×3) = 2 ×3 2 Proof: 2 ×3 = 4 ×9 = 36 2 2 2 2 2
  20. 20. Power of zero • Any nonzero number raised to the power of zero is one. a =1 0
  21. 21. Quotient of powers • To find the quotient of powers with the same base, subtract the exponents. a =a n a m m-n
  22. 22. •For example… 3 ÷3 =3 2 2 2-2 3 = 3 • 3 =9 =1 2 3 3•3 2 9 =3 0
  23. 23. Power of a quotient To raise a quotient to a power, raise both the numerator and denominator to that power. (a/b) = a /b m m m
  24. 24. 3 3 2 2 Example:  ÷ = 3 7 7
  25. 25. Reciprocals • To change a sign of an exponent, move the expression to the denominator of a fraction, or to the numerator. a = 1/a -n n 1/a = a -n n
  26. 26. 1 1 Example #1: 2 = 3 = 2 8 −3 3 1 5 3 Example #2: − 3 = = 5 = 125 5 1
  27. 27. references Maslijr, 2013, Laws of exponents, viewed 05 March 2014, from http://www.slideshare.net/masljr/laws-of-exponents-23863798 Melnichenko, Y., 2008, Exponents, viewed 05 march 2014, from http://www.slideshare.net/yelena585/exponents-presentation?qid=c977ae81-f7bd48bf-bf86-c88dd4657047&v=default&b=&from_search=30 Morris, B., 2012, Grade 6 exponents lesson, viewed 05 March 2014, from http://www.slideshare.net/BobMorris72/math-exponents?qid=c977ae81-f7bd48bf-bf86-c88dd4657047&v=default&b=&from_search=2 Scallion, K., 2010, Rules of exponents, viewed 05 March 2014, from http://www.slideshare.net/kscallion/rules-of-exponents Wilkerosn,K., 2013, Exponents, viewed 05 march 2014, from http://www.slideshare.net/katiewilkerosn/exponents-27460394?qid=c977ae81f7bd-48bf-bf86-c88dd4657047&v=default&b=&from_search=11
  28. 28. THE END
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