2. EXPONENTS
• A quantity representing the power to which
a given number or expression is to be
raised, usually expressed as a raised
symbol beside the number or expression
(e.g. 3 in 23 = 2 × 2 × 2).
3. General Enquiry
• Exponents are shorthand for repeated
multiplication of the same thing by itself.
For instance, the shorthand for multiplying
three copies of the number 5 is shown on
the right-hand side of the "equals" sign
in (5)(5)(5) =53. The "exponent", being 3 in
this example, stands for however many
times the value is being multiplied. The
thing that's being multiplied, being5 in this
example, is called the "base".
4. Exponents
35 Power
exponent
base
3 3 means that is the exponential
Example:
form of t
125 5 5
he number
125.
53 means 3 factors of 5 or 5 x 5 x 5
5. The Laws of Exponent
Comes From 3 ideas
• The exponent says how many times to
use the number in a multiplication.
• A negative exponent means divide,
because the opposite of multiplying is
dividing
• A fractional exponent like 1/n means
to take the nth root:
7. The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
n
x x x x x x x
x
n
times
n factors of x
Example: 53 555
8. #2: Multiplying Powers: If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
m n m n x x x
So, I get it!
When you
multiply
Powers, you
add the
exponents!
2 6 2 3 2 6
3 29
512
9. #3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
m n m n
n
x
x x x
x
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
6
2 6 2 4
2 2
16
2
2
10. #4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
n
xm xmn
So, when I
take a Power
to a power, I
multiply the
exponents
3 2 3 2 5 (5 ) 5 5
11. #5: Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result is a
multiplication of individual factors of the product, each powered
by the given exponent.
n xy xn yn
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
2 2 2 (ab) a b
12. #6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
n n
x x
y y
n
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
16
81
4 2
4
3
2
3
4
13. Try these:
2 5 1. 3
3 4 2. a
3. 2a
2 3 4. 2 2 a 5 2 b
3
5. ( 3a 2 )
2 6. 2 3 s t
4
5
7.
s
t
2
9
3
5
3
8.
2
8
4
9.
st
rt
2
5 8
a b
4 5
36
4
10.
a b
14. SOLUTIONS
2 5 1. 3
3 4 2. a
2 3 3. 2a
2 5 3 2 4. 2 a b
2 2 5. ( 3a )
2 4 3 6. s t
10 3
12a
3 2 3 6 2 a 8a
2 2 5 2 3 2 4 10 6 10 6 2 a b 2 a b 16a b
2 2 2 4 3 a 9a
2 3 4 3 6 12 s t s t
15. SOLUTIONS
5
7.
s
t
2
9
3
5
3
8.
2
8
4
9.
st
rt
2
5 8
a b
4 5
36
4
10
a b
2 8
5
5
s
t
4 2 8 3 3
s t
2
2
4
r
st
r
3 2 2 2 3 2 2 6 9ab 9 a b 81a b
16. #7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
1 m
m x
x
So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
1
1
3 9
and
1
3
125
5
5
2
2
3
3
17. #8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
0 1 x
0
5 1
1
and
0
0
(5 )
1
a
a
and
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.
18. Try these:
2 0 1. 2a b
2 4 2. y y
5 1 3. a
4. s 2 4s
7 2 4 5. 3x y
3
2 4 0 6. s t
19. SOLUTIONS
2 0 1. 2a b
5 1 3. a
1
a
2 7 4. s 4s
2 3 4 5. 3x y
2 4 0 6. s t
1
5
4s
5 x
8
4 8 12
12
81
3
y
x y
1