What are Exponents?
Exponents are used to show repeated multiplication of a
number by itself.
For example, 7 × 7 × 7 can be represented as 73.
• the exponent is ‘3’ which stands for the number of times
the number 7 is multiplied.
• 7 is the base here which is the actual number that is
getting multiplied.
What are Exponents?
So basically exponents or powers denotes the number of times
a number can be multiplied.
If the power is 2, that means the base number is multiplied two
times with itself.
Some of the examples are:
34 = 3×3×3×3
105 = 10×10×10×10×10
z3 = z × z × z
a1 = a
a2 = a x a
a3 = a x a x a
a4 = a x a x a x a
.
.
.
a7 = a x a x a x a x a x a x a
… and so on !
Check your Understanding
Assignment - Complete:
• WKS #1 Introducing Exponents 1
• WKS #2 Expanded Form to Exponential Form
Turn in completed worksheets in the box
Lesson #2 Exponent Rules
Rules of Exponents With Examples
Suppose, a number ‘a’ is multiplied by itself
n-times, then it is represented as an where a
is the base and n is the exponent.
Exponents follow certain rules that help in
simplifying expressions which are also called
its laws. There are different laws or rules
defined for exponents. The important laws of
exponents are shown at the right:
Power of One
An exponent tells you how many times the base number is used as a
factor. ... Any number raised to the power of one equals the number
itself, such as;
a1 = a
where ‘a’ is any non-zero term.
Example: What is the value of 71 – 31 ?
Solution: 71 – 31 = 7 – 3 = 4
Power of Zero
According to this rule, when the power of any integer is zero, then its
value is equal to 1, such as;
a0 = 1
where ‘a’ is any non-zero term.
Example: What is the value of 50 + 22 + 40 + 71 – 31 ?
Solution: 50 + 22 + 40 + 71 – 31 = 1 + 4 + 1 + 7 – 3 = 10
Negative Exponent Rule
According to this rule, if the exponent is negative,
we can change the exponent into positive by
writing the same value in the denominator and the
numerator holds the value 1.
Example: Find the value of 2-2
Solution: 2-2 can be written as 1/22 = 1/4
In other words, we can say that, if “a” is a non-zero number or non-zero
rational number, we can say that a-m is the reciprocal of am.
a0 = 1
Check your
Understanding
Assignment - Complete:
• WKS #3 Introduction to Exponents
0
a =
1
1
a =
a
Turn in completed worksheets in the box
Lesson #3 Exponent Rules
a0 = 1
a1 = a
a2 = a x a
a3 = a x a x a
a4 = a x a x a x a
… and so on !
Check your Understanding
Product With the Same Bases
As per this law, for any non-zero term a,
am × an = am+n
where m and n are real numbers.
Example: What is the simplification of 55 × 51 ?
Solution: 55 × 51 = 55+1 = 56
Example: What is the simplification of (−6)-4 × (−6)-7?
Solution: (−6)-4 × (−6)-7 = (-6)-4-7 = (-6)-11
Note: We can state that the law is applicable for negative terms also.
Therefore the term m and n can be any integer.
Use the Exponent Product Rule and
answer in a single exponent.
am × an = am+n
Check your Understanding
Use the Exponent
Product Rule and
answer in a single
exponent.
Quotient with Same Bases
As per this rule,
am / an = am-n
where a is a non-zero term and m and n are integers.
Example: Find the value when 10-5 is divided by 10-3.
Solution: 10-5 / 10-3 = 10-5-(-)3 = 10-5+3 = 10-2 = 1 / 100
Use the Exponent Quotient Rule and
answer in a single exponent.
am / an = am-n
Check your Understanding
Use the Exponent
Quotient Rule and
answer in a single
exponent.
Assignment - Complete:
• WKS #
Turn in completed worksheets in the box
Lesson #4 Exponent Rules
Power Raised to a Power
According to this law, if ‘a’ is the base, then the power raised to the
power of base ‘a’ gives the product of the powers raised to the base
‘a’, such as;
(am)n = amn
where a is a non-zero term and m and n are integers.
Example: Simplify and write the exponential (23)3
Solution:
(23)3 = 23x3 = 29
(am)n = amn
Check your Understanding
Product to a Power
As per this rule, for two or more different bases,
if the power is same, then;
an bn = ( ab )n
where a is a non-zero term and n is the integer.
Example 5: Simplify and write the exponential 2-3 x 5-3
Solution:
2-3 x 5-3 = (2 × 5)-3 = 10-3
an bn = ( ab )n
1)
52 • 32
2)
93 • 43
3)
37 • 87
4)
a11 • b11
5)
x5 • y5
6)
71 • 111
Check your Understanding
1)
44 • 9 4
2)
26 • 5 6
3)
122 • 32
4)
615 • v15
5)
w5 • y5
6)
r9 • 29
7)
k0 • 110
8)
x3 • y3
Assignment - Complete:
• WKS #
Turn in completed worksheets in the box
Lesson #5 Exponent Rules
Quotient to a Power
As per this law, the fraction of two different bases with the same
power is represented as;
an / bn = (a / b)n
where a and b are non-zero terms and n is an integer.
Example: Simplify the expression and find the value: 153 / 53
Solution:
153 / 53 = ( 15 / 5 )3 = 33 = 27
n
a /
n
b =
(a /
n
b)
Check your Understanding
Fractional Exponent Rule
The fractional exponent rule is used, if the exponent is
in the fractional form.
Here, a is called the base, and 1/n is the exponent,
which is in the fractional form.
Thus, a1/n is said to be the nth root of a.
Example: Simplify: 41/2
Solution: According to the fractional exponent rule,
41/2 can be written as √4
41/2 = √4 = 2 (As, the square root of 4 is 2)
Fractional Exponent Rule
Check your Understanding
Assignment - Complete:
• WKS #
Turn in completed worksheets in the box