Download Properties of Exponents

Core Plus Mathematics IV
Properties of Exponents Worksheet
Name_____________________
Per. _______
Properties of Exponents
I’m the exponent.
I’m up high and
smaller
Definition of Natural Number Exponents:
a  a  a  ...  a  a  a n
n factors of a
a c a

bc b
Factor Cancellation Property of Fractions:
Fraction Multiplication:
I’m the base
Together we
form an
exponential
expression
a c ac
 
b d bd
Product Rule of Exponents
a 2a 3
=
=
b 4b10
=
a na m
=
Use the definition of exponents to rewrite the expression as
a product without exponents
Now use the definition of exponents to rewrite the product
as an exponential expression (an expression with just a base
and one exponent)
Rewrite this expression as an exponential expression with
just one exponent
Generalize your result and rewrite this expression as single
exponential expression
Power Rule of Exponents
(a 2 )3 =
=
=
( c 4 )7
Use the definition of exponents to rewrite the expression
inside of parenthesis as a product without exponents
Use the definition of exponents to rewrite the previous
expression without exponents
Now use the definition of exponents to rewrite the product
as single exponential expression (an expression with just a
base and one exponent)
Rewrite this expression as single exponential expression
=
(a n )m =
Generalize your result and rewrite this expression as single
exponential expression
Power of a Product Rule
(ab)3 =
=
=
 cd 
5
=
(ab)n =
Use the definition of exponents to rewrite the expression as
a product without exponents
Now use the commutative and associative properties of
multiplication to rewrite this product so that the a factors
are together and the b factors are together
Rewrite the expression so that the factor a is raised to an
exponent and the factor b is raised to an exponent
Rewrite the expression as a product of factors raised to
exponents
Generalize your result and rewrite the expression as a
product of factors raised to exponents.
Power of a Quotient Rule
a
 
b
4
=
Use the definition of exponents to rewrite this expression as
a product without exponents.
=
Now use the definition of fraction multiplication to rewrite
the expression as a single fraction without exponents
Rewrite the expression as single fraction with an
exponential expression in the numerator and an exponential
expression in the denominator
=
x
 y
 
a
 
b
=
Rewrite the expression as single fraction with an
exponential expression in the numerator and an exponential
expression in the denominator
=
Generalize your result and rewrite the expression as single
fraction with an exponential expression in the numerator
and an exponential expression in the denominator.
8
n
Quotient Rule
a7
a3
=
=
=
Use the definition of exponents to rewrite this expression as
a single fraction without exponents in the numerator or
denominator
Use the factor cancellation property of fractions to reduce
this fraction.
Rewrite the expression as a single exponential expression
(no fractions)
x15
x7
=
Rewrite the expression as a single exponential expression
(no fractions)
an
am
=
Generalize your result and rewrite the expression as a single
exponential expression (no fractions)
Extending the Definition of Exponents
We have defined exponents according to this definition
x  x  x  ...  x  x  x n
n factors of x
But using this definition restricts us to the natural numbers = {1, 2, 3, ….}. After all, what would it mean to have zero factors of x?
Or – 2 factors of x? Or ¾ factors of x?
Whenever mathematicians define a symbol or concept on a subset of the real numbers, they like to extend it to bigger sets of the real
numbers, and if possible to the whole real number system.
So let’s try to extend the definition of exponents to the whole numbers = {0, 1, 2, 3, ….}. The only number in the whole numbers that
is not in the natural numbers is the number zero.
When mathematicians extend the definition of a concept they do so that all the properties that were true on the smaller set of numbers
are also true on the larger set of numbers. We’ll extend the definition of exponents to include zero in such a way that the quotient rule
is still true.
Exponent of Zero
23
23
=
=
an
an
=
an
an
=
Rewrite the numerator and denominator of the fraction as a
whole number using the definition of exponents
By definition of fractions, what whole number does the
previous fraction equal.
In general what whole number does this fraction equal (if
a  0)
Now use the quotient rule from above to write this
expression as a single base that’s raised to an exponent that
is the difference of two values
Rewrite the expression as a base raised to a whole number
=
If two expressions are equal to the same thing, then they
must equal each other. Given the two expressions above
a0
=
that are equal to
an
0
, what can we conclude about a ?
an
Negative Integer Exponents
Now that we have extended the definition of exponents to whole numbers, let’s see if we can extend the definition to the integers =
{…-2, -1, 0, 1, 2, 3, …}. Of course the integers contain the whole numbers, so we just have to extend the definition to the negative
integers.
Again we will extend the definition in such a way that the quotient rule is still true.
a3
a5
Rewrite the numerator and denominator of the fraction as
products using the definition of exponents
=
=
a3
a5
Use the cancellation property of fractions to reduce the
fraction.
Now use the quotient rule to rewrite this expression as the
base raised to the difference of two numbers
=
Rewrite the expression as a base raised to an integer.
=
We have two expressions equal to
=
a3
, therefore they must
a5
equal each other. Set these two expressions equal to each
other.
Generalize your result to a
a n
n
=
Rational Number Exponents
Now let’s extend the definition to rational numbers. The rational numbers remember are numbers that can be written as p/q where p
and q are both integers. All integers are also rational numbers since an integer can always be expressed as a fraction with a
denominator of one (eg. 7 = 7/1). So we only have to extend the definition to rational numbers that are not integers such as 2/3.
But before we do this, we will extend the definition just to rational numbers that have a numerator of one, such as 1/3 or 1/5.
1
1
1
2 32 32 3
=
=
=
Use the product rule to rewrite this expression as a base
raised to the sum of 3 numbers
Now rewrite the expression as a base raised to one number
Now rewrite the expression as a whole number not raised to
any base.
Rational Number Exponents (Continued)
1
3
2
a
1
3
1
n
So 2 is a number that when I multiply it by itself three
times equals 2. Can you think of another notation to use to
represent a number that when we multiply it by itself three
times equals two? Think how we would represent a number
that when multiplied by itself equals 2.
=
Generalize your result and define an exponent of the form
1/n.
=
Now we want to define exponents for any type of rational number, not just those that have a numerator equal to one.
2
a3
=
=
=
a
m
n
Rewrite the expression as an exponential expression with an
exponent that is the product of two fractions (Hint: One
fraction should have a numerator of one and the other
fraction should have a denominator of one)
Use the power rule of exponents to rewrite the expression as
an exponential expression in parenthesis and an exponent
outside of parenthesis such that the exponent inside
parenthesis has a numerator of 1.
Now rewrite the expression inside parenthesis in radical
notation
Generalize your result and define an exponent of the form
m/n.
=
2
Evaluate
8 3 without a calculator using the definition above.
2
83
=
=
=
Rewrite the expression as an exponential expression with an
exponent that is the product of two fractions
Use the power rule of exponents to rewrite the expression as
an exponential expression in parenthesis and an exponent
outside of parenthesis with a denominator of one
Now rewrite the expression inside parenthesis in radical
notation
Evaluate the radical inside parenthesis
=
=
Raise the number inside parenthesis to the exponent outside
the parenthesis
We now know how to evaluate exponents for any rational number.
Question:
Answer: (Use a complete sentence)
For what type of real
number do we not
know how to
evaluate exponents?
Special Fraction Rule
a
 
b
n
Rewrite the expression using the rule for negative integer
exponents
=
=
=

Use the exponent rule for fractions to rewrite the
denominator of the previous expression as fraction with an
exponential expression in the denominator and the
numerator.
Rewrite the expression as the division of two fractions using
a division symbol
Divide the two fractions according to the rule for fraction
division
Rewrite the expression as a fraction raised to an exponent
using the fraction property of exponents