EVALUATING
EXPONENTS
Let’s review the odd/even rule
with and without ( ):
EVEN POWERS:
(2)      2  2  2  2  16
4
2 
4
 2  2  2  2  16
In this first one, BOTH the
negative sign and the 2 are to the
4th power so there are 4 negatives (even
number so positive).
In the second one, ONLY the 2 is to the
4th power because there are no ( )…I call
the negative sign a ZAPPER!
AFTER you simplify the power, then think
of the negative sign as multiplying by -1. In
order of operations, you do the power first
and multiply by -1 AFTER you simplify the
power.
Let’s review the odd/even rule
with and without ( ):
ODD POWERS:
( 2)    2  2  2  8
3
2 
3
2  2  2
 8
So for different reasons,
negative bases raised to
an ODD power are ALWAYS
NEGATIVE!
In the first one, BOTH the
negative sign and the 2 are to the
3rd power…so there are 3 negatives, which
is an odd number. The answer is
NEGATIVE.
In the second one, ONLY the 2 is to the
3rd power…the negative sign is a ZAPPER.
AFTER you simplify the power, you
multiply by -1. There is only 1 negative in
this problem and 1 is an odd number so
the answer is NEGATIVE.
So now you try these:
(2)  64
6 negatives is POSITIVE
2  32
1 negative is NEGATIVE
6
5
2  64
6
There is ONLY 1 NEGATIVE! WHY?
When there are no ( ), only the base
is raised to the power and the negative sign
is the same as multiplying by a -1 AFTER the power
is simplified…therefore the answer is NEGATIVE
(2)  32
5
There are 5 negatives. Why?
When the negative is inside the ( ), it’s raised to the power of 5.
An odd number of negatives is NEGATIVE
TRY THIS:
(2)  64
6
How many negatives are there in this problem?
7 (which is odd).
There are 6 negatives from the power and another negative (a zapper)
outside the ( )
(2)  32
5
How many negatives are there in this problem?
6 (which is even).
There are 5 negatives from the power and another negative (a zapper)
outside the ( )
Try this:
16(9  11)

4
2
3
DO NOT DISTRIBUTE THE 16
BECAUSE ORDER OF OPERATIONS TELLS YOU
THAT YOU MUST SIMPLIFY THE POWER
(squared) BEFORE YOU CAN MULTIPLY!
Do what’s in the ( ) first:
16(9  11) 16(2)


4
4
2
3
2
3
Now simplify the exponents:
16(2) 16(4)


4
64
2
3
Simplify the numerator:
16(4) 64


64 64
Once the numerator and denominator are simplified, simplify the fraction:
64
 1
64
TRY THIS:
(30  3  5)  (25) 
2
2
DO NOT DISTRIBUTE THE FIRST NEGATIVE SIGN
UNTIL
AFTER YOU SIMPLIFY
WHAT’S INSIDE THE PARENTHESES
AND
AFTER YOU SIMPLIFY THE OUTSIDE EXPONENT.
Do the exponent inside the ( ) and ++ for the 25:
(30  3  5)  (25)  (30  9  5)  25
2
2
2
Keep simplifying in the ( ):
(30  9  5)  25  (30  45)  25
2
2
Keep simplifying in the ( ):
(30  45)  25  (15)  25
2
2
Now do the power:
(15)  25  (225)  25
2
Add:
(225)  25  200
TRY THIS:
x  3 y  4
( x  2) ( y  2)
2
3
REMEMBER:
You always plug in negative
numbers with ( )
x  3 y  4
( x  2) ( y  2)
2
3
[(3)  2] [(4)  2]
2
3
[3  2] [6]
2
(1) (6)
2
3
3
(1)(216)
216
You always plug in negative
numbers with ( )!
You can go + + on the 3 because the
power is on the OUTSIDE.
Simplify the ( )
Simplify the exponents
Multiply
TRY THIS:
x  3 y  4
 xy
2
Only the y is raised to the 2nd power
x  3 y  4
 xy
2
(3)(4)
(3)(4)
3(16)
48
2
2
Plug in negative bases in ( )
You can do ++ on the 3 because it’s not raised to a power
The -4 is squared so it is +16 because it’s an even power
Multiply
TRY THIS:
x  3 y  4
 x  x  x  x  4 x  (9)
5
4
3
2
Make sure you always simplify the EXPONENT 1st, then apply the negative
sign given in the problem…In other words, YOU CAN’T GO + + BEFORE
YOU DO THE EXPONENT INSIDE THE ( )!!!
x  3 y  4
 x  x  x  x  4 x  (9)
5
4
2
3
Plug in the negative base in ( ):
(3)  (3)  (3)  (3)  4(3)  (9)
5
4
3
2
Simplify the exponents:
(243)  (81)  (27)  (9)  12  9
Simplify the negative signs and add:
243  81  27  9  12  9  219
TRY THIS:
x  5 and y  3
( x  y)

x y
2
2
2
x  5 and y  3
( x  y)
x y
2
(5  (3))
(5)  (3)
2
2
2
2
(5  3)
25  (9)
(8)
16
2
2
64
4
16
2
Plug in the negative base in ( )
Numerator: Simplify inside the ( )
Denominator: Simplify the exponents
Numerator: Simplify inside the ( )
Denominator: Add
Numerator: Simplify the exponents
Finally: Simplify the fraction