1. Exponents

2. Negative Exponents
Key idea: turn the exponent into a positive, then simplify.
Opposite place, opposite sign.
Turn the exponent into a fraction.
Next, flip.
When you flip, the negative exponent becomes positive.
Example: 9⁻²  9⁻²/1  1/9²  1/81
When you have an unknown, it cannot have a value that will cause the denominator to be zero.
We can’t divide by zero.

3. Multiplying Numbers with the Same Base
If a number is multiplied by another number with the same base, you add the exponents.
Example:
2³ x 2⁴ = (2 x 2 x 2)(2 x 2 x 2 x 2) = 2³⁺⁴ = 2⁷ = 128
2³ x 2⁴ = (8)(16) = 128

4. Dividing Numbers with the Same Base
If a number is divided by another number with the same base, you can subtract the exponents.
Example:
2⁵ ÷ 2² = 2⁵¯² = 2³ = 8

5. Zero as an Exponent Anything with a zero exponent equals 1. Examples:
10⁰ = 1
10² ÷ 10² = 10²¯² = 10⁰ = 1

6. Raising a Power to a Power
When raising an exponent to another exponent, you fast add or multiply the exponents.
Example:
(6⁷)³ = 6⁷˚³ = 6^21

7. Simplifying an Exponential Expression
There are two techniques that can be used to simplify an exponential expression.
The first is to have everything in the numerator. (# on top)
The second is to make all the exponents positive.

8. Examples: Put everything in the numerator: X³X¯² = X³X¯²X¯²X¯⁴ = X¯⁵
Make all the exponents positive:
X³X¯² = X³ = X³ = X¯⁵ = 1
X²X⁴ X²X⁴X² X⁸ X⁵