Transcript Exponents

Exponents
Positive, Negative,
and Square Roots
Vocabulary
Power
Definition:
Example:
• A number that can
be written using an
exponent
• The power 7² is
read as seven to the
second power or
seven squared
Factor
Definition:
Example:
• A number that
divides into a whole
number with a
remainder of zero.
• The factors of 12
are 1, 2, 3, 4, 6, and
12.
Base
Definition:
Example:
• In a power, the
number used as a
factor.
• In 6³, the base is 6.
That is, 6³ = 6 ∙ 6 ∙
6.
Exponent
Definition:
Example:
• In a power, the
number of times the
base is used as a
factor.
• In 5³, the exponent
is 3. That is, 5³ = 5
∙ 5 ∙ 5.
Squared
Definition:
Example:
• A number raised to
the second power.
• Five squared is
written as 5², which
means 5 ∙ 5.
Cubed
Definition:
Example:
• A number raised to
the third power
• Nine cubed is
written as 9³, which
means 9 ∙ 9 ∙ 9.
Examples
Write each of the following statements using
exponents:
• Three to the fourth power
• Twelve squared
• Nine cubed
Examples
Write each of the following using exponents:
• 3 ∙ 3 ∙3 ∙ 3
• 2∙2∙2∙2∙2∙2∙2∙2
• x∙x∙x∙x∙x∙x
Examples
Write each of the following in expanded
form:
76
54
10 2
Examples
Evaluate each of the following:
82
10 3
25
Zero
Note:
Examples:
• Anything to the zero
power is equal to 1.
•
•
•
•
•
5º = 1
15º = 1
2,342º = 1
xº = 1
yº = 1
One
Note:
Examples:
• Any number raised
to the first power is
that number.
• 2¹ = 2
• 56¹ = 56
• x¹ = x
Some Quick Review
• Multiplicative Inverse: a number times its
multiplicative inverse is 1 (the identity).
• Another name for multiplicative inverse is
reciprocal.
• Examples:
– The multiplicative inverse of
– The multiplicative inverse of
3
4
7
9
is
is
4
3
9
7
.
.
Negative Exponents
• Example:
x
2
1
 2
x
• Explanation:
Since exponents are another way to write
multiplication and the negative is in the exponent, to
write it as a positive exponent we do the
multiplicative inverse which is to take the reciprocal
of the base.
Negative Exponents/Reciprocals
Examples
Rewrite each of the following in standard form.
8 2
2 5
9 3
7 4
Multiplying Powers
Product of Powers:
Product of Powers:
In words:
In symbols:
• You can multiply
powers that have
the same base by
adding the
exponents.
• For any number a
and positive integers
m and n,
a a  a
m
n
m n
Examples
5 5  55555  5
2
3
2 3
5
5
8 8  888888888  8
3
6
3 6
x x  x x x x x  x
2
3
2 3
8
x
• Note: This only works if the bases are the
same.
9
5
Practice
1. (3a²)(4a³)
2. d ∙ d³ ∙ d
3. m³(m³n)
4. (x²y²)(x³y³)
Dividing Powers
Quotient of Powers:
Quotient of Powers:
In words:
In symbols:
• You can divide
powers that have
the same base by
subtracting the
exponents.
• For any whole
numbers m and n,
and nonzero number
a,
m
a
m n
a
n
a
Examples
54 5  5  5  5
4 2
2


5

5
52
55
78 7  7  7  7  7  7  7  7
8 5
3


7

7
75
7 7 7 7 7
64
6666
1
4 7
3

6
6  3
7
6
6666666
6
x8 x x x x x x x x
85
3


x

x
x5
x x x x x
• Note: This only works for powers that have
the same base.
Practice
1. 8³
8²
4. x³y
x²y
2. 9²
9²
5. m²n³
mn³
3. x³
x
6. 45x²
15x
Some basics that make
exponents easier to remember
• Know your multiplication facts:
– You should have the multiplication table from 1 x
1 to 12 x 12 committed to memory.
– That means you should have them memorized
and not have to use your fingers to figure them
out.
• If you need to work on your multiplication facts,
you can make flashcards to help you practice.
• You can even work on filling out tables as practice.
Squares
• You need to
memorize the
products of 1
through 15
squared.
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
Cubes
• You need to
memorize the
products of 1
through 10
cubed.
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1.000
Explore
Simplify each of the following expressions: (Hint: write the powers out.)
1. (-5)³
2. (2x)²
3. (-3xy)³
4. (7abc)4
5. (2x²)³
6. (5x³y²)³
7. (2x³y³z)6
Raising a Power to a Power
Property:
• When raising a power to a power multiply the
exponents.
Explanation:
• When an expression that contains and exponent
is written in parentheses and the parentheses
have an exponent, multiply the exponents to
find the power of the simplified expression
Examples
1.
(4v²)² = (4v²)(4v²) = 16v4
2. (2x³y³)³ = (2x³y³)(2x³y³)(2x³y³) = 8x9y9
3. (-6xy4)5 = (-6xy4)(-6xy4)(-6xy4)(-6xy4)(-6xy4) = -7776x5y20
4. (-2a4b5c7)8 = (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7)
(-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7)
= -256a32b40c56
•
Do not copy this sentence, but there is a mistake in one of the
examples, first to spot it and raise their hand to correct it gets a
Hershey kiss :-)
Square Root
• Remember: To square a number means
to multiply the numbers by itself.
• When you find the square root of a
number you are looking for the factor
that when multiplied by itself gave you
the number.
Square Root (cont.)
Definition:
• One of two equal
factors of a number
• This symbol refers to
the square root of a
number and is called a
radical sign.
Example:
1 1
4 2
9 3
16  4
25  5
Perfect Squares
• Not all numbers have a whole number as
a square root.
• Numbers that have a whole number as a
square root are referred to as perfect
squares.
Perfect Squares (cont.)
The first fifteen perfect squares are:
• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
144, 169, 196, and 225.