21. Common Mistake
-24 ≠(does not equal)(-2)4
Without the parenthesis, positive 2 is
multiplied by itself 4 times; then the answer
is negative.
With the parenthesis, negative 2 is
multiplied by itself 4 times; then the answer
becomes positive.
22. Common mistake
-24 = (-1)x(x means
times)
+24
=
Why?
The 1 and the positive sign are invisible.
Anything x 1=anything, so 1 x 2 x 2 x 2 x 2 = 16;
and negative x positive = negative
23. Common Mistake
(-2)4=- 2 x -2 x -2 x -2 = +16
Why?
Multiply the numbers: 2 x 2 x 2 x 2 = 16 and
then multiply the signs:
1st negative x 2nd negative = positive; that
positive x 3rd negative = negative;
that negative x 4th negative = positive; so
answer = positive 16
24.
25. The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
n
n times
x x x x x x x x
3
Example: 5 5 5 5
n factors of x
26. #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
512
2
2
2
2 9
3
6
3
6
So, I get it!
When you
multiply
Powers, you
add the
exponents!
27. #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!
16
2
2
2
2 4
2
6
2
6
28. Try these:
2
2
3
3
.
1
4
2
5
5
.
2
2
5
.
3 a
a
7
2
4
2
.
4 s
s
3
2
)
3
(
)
3
(
.
5
3
7
4
2
.
6 t
s
t
s
4
12
.
7
s
s
5
9
3
3
.
8
4
4
8
12
.
9
t
s
t
s
5
4
8
5
4
36
.
10
b
a
b
a
29.
2
2
3
3
.
1
4
2
5
5
.
2
2
5
.
3 a
a
7
2
4
2
.
4 s
s
3
2
)
3
(
)
3
(
.
5
3
7
4
2
.
6 t
s
t
s
81
3
3 4
2
2
7
2
5
a
a
9
7
2
8
4
2 s
s
SOLUTIONS
6
4
2
5
5
243
)
3
(
)
3
( 5
3
2
7
9
3
4
7
2
t
s
t
s
31. #4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
n
m mn
x x
So, when I
take a Power
to a power, I
multiply the
exponents
5
2
3
2
3
5
5
)
5
(
32. #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 n n
xy x y
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
2
2
2
)
( b
a
ab
33. #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
n
x x
y y
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
81
16
3
2
3
2
4
4
4
34. Try these:
5
2
3
.
1
4
3
.
2 a
3
2
2
.
3 a
2
3
5
2
2
.
4 b
a
2
2
)
3
(
.
5 a
3
4
2
.
6 t
s
5
.
7
t
s
2
5
9
3
3
.
8
2
4
8
.
9
rt
st
2
5
4
8
5
4
36
.
10
b
a
b
a
35.
5
2
3
.
1
4
3
.
2 a
3
2
2
.
3 a
2
3
5
2
2
.
4 b
a
2
2
)
3
(
.
5 a
3
4
2
.
6 t
s
SOLUTIONS
10
3
12
a
6
3
2
3
8
2 a
a
6
10
6
10
4
2
3
2
5
2
2
16
2
2 b
a
b
a
b
a
4
2
2
2
9
3 a
a
12
6
3
4
3
2
t
s
t
s
37. #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!
9
3
3
1
125
1
5
1
5
2
2
3
3
and
38. #8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
0
1
x
1
)
5
(
1
1
5
0
0
0
a
and
a
and
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.
39. Try these:
0
2
2
.
1 b
a
4
2
.
2 y
y
1
5
.
3 a
7
2
4
.
4 s
s
4
3
2
3
.
5 y
x
0
4
2
.
6 t
s
1
2
2
.
7
x
2
5
9
3
3
.
8
2
4
4
2
2
.
9
t
s
t
s
2
5
4
5
4
36
.
10
b
a
a
40. SOLUTIONS
0
2
2
.
1 b
a
1
5
.
3 a
7
2
4
.
4 s
s
4
3
2
3
.
5 y
x
0
4
2
.
6 t
s
1
5
1
a
5
4s
12
8
12
8
4
81
3
y
x
y
x
1
