1. . . . are the mathematician’s shorthand.
Exponents . . .
. . . are the mathematician’s shorthand.

2. The Basics

3. definitions In general, the format for using exponents is:
where the exponent tells you how many of the base are being multiplied together.
Exponents are also referred to as "powers".
For example, 23 can be read as "two cubed" or as "two raised to the third power".

4. exponents of negative values
-36 is NOT the same as (-3)6!
When we multiply negative numbers together, we must use parentheses to switch to exponent notation.

5. even and odd powers
Even powers of negative numbers allow for the negative values to be arranged in pairs.  This pairing guarantees that the answer will always be positive.  
(-5)6 = (-5)•(-5)  •   (-5)•(-5)   •   (-5)•(-5)   ← All pairs.
=     25       •       25       •       25   =  15625   (a positive answer)
Odd powers of negative numbers, however, always leave one factor of the negative number not paired.  This one lone negative term guarantees that the answer will always be negative.
(-5)5 = (-5)•(-5)  •   (-5)•(-5)   •   (-5)   ← One lone, un-paired, negative.  
=     25       •       25       •   (-5)   =  -3125   (a negative answer)

6. zero exponents
The value of any expression* raised to the zero power is 1.
*Zero raised to the zero
power is undefined.
baseexponent
value 20
(-6)0 40
-80 00 1 1 1 1
undefined

7. negative exponents 4-1 (-5)-2 7-3
Negative numbers as exponents have a special meaning:
negative exponent
positive exponent
4-1
(-5)-2
7-3

8. Rewrite the following expression without using exponents: 84
PRACTICE
Rewrite the following expression without using exponents:    84
4 • 4 • 4 • 4 • 4 • 4 • 4 • 4
8 • 8 • 8 • 8
8 • 4

9. Which of the following is equal to the expression 3-7 ?
PRACTICE
Which of the following is equal to the expression 3-7 ? 37
(-3)7

10. What is the value of the expression (-2)3(3)2 ?
PRACTICE
What is the value of the expression
(-2)3(3)2 ?
-72 72
-36

11. Which of the following has a value of 1?
PRACTICE
Which of the following has a value of 1?
(-1)2
1-1 10
all of the above

12. The expression (30)(40)(22) is equal to:
PRACTICE
The expression  (30)(40)(22) is equal to: 4 1
none of the above

13. Exponent Rules

14. . . . when in doubt, expand terms to see what is happening.
Multiplying Powers
Rule #1:
For all numbers x and all integers m and n,
This simply means...
when you are multiplying, and the bases are the same, you ADD the exponents.
x2 • x3 = (x • x) • (x • x • x)
= x5
. . . when in doubt, expand terms to see what is happening.

15. . . . when in doubt, expand terms to see what is happening.
Dividing Powers
Rule #2:
For all numbers x (not zero)
and all integers m and n,
This simply means...
when you are dividing, and the bases are the same, you SUBTRACT the exponents.
. . . when in doubt, expand terms to see what is happening.

16. Power to a Power Rule #3: For all numbers x and all integers m and n,
This simply means...
when you raise a power to a power, you MULTIPLY the exponents.
. . . when in doubt, expand terms to see what is happening.

17. Power of Products Rule #4: For all numbers x and y and all integers n,
This simply means...
each factor of the product gets raised to the new power.
. . . when in doubt, expand terms to see what is happening.