1. Product Rule (Multiplying Powers with the Same Base)
When multiplying terms with the same base, you add the exponents.
2. Quotient Rule (Dividing Powers with the Same Base)
When dividing terms with the same base, you subtract the exponents.
3. Power of a Power Rule
To raise a power to another power, you multiply the exponents.
4. Power of a Product Rule
The power of a product is equal to the product of the powers.
5. Power of a Quotient Rule
The power of a quotient is equal to the quotient of the powers.
These rules handle zero, negative exponents, and negative bases.
6. Zero Exponent Rule
Any non-zero number raised to the power of zero is equal to 1.
Note: $0^0$ is undefined.
7. Negative Exponent Rule
A term raised to a negative exponent is equal to its reciprocal with a positive exponent.
8. Negative Exponent of a Fraction
To remove a negative exponent on a fraction, flip the fraction (take the reciprocal) and make the exponent positive.
9. Negative Base with Exponent
$(-1)^{\text{even number}} = 1$
An even power of -1 results in 1; an odd power results in -1.
10. General Rule for Negative Base
$(-a)^n = a^n \text{, if } n \text{ is even}$
If the base is negative, an even exponent makes the result positive, while an odd exponent keeps the result negative.
These formulas connect exponents with radicals (roots).
11. Equality of Exponents (Same Base)
If two powers are equal and have the same base (that is not -1, 0, or 1), then their exponents must be equal.
12. Nth Root Rule
The $n$-th root of a number can be written as the number raised to the power of $\frac{1}{n}$.
13. Fractional Exponent Rule