Properties_of_Exponents - Product & Power Rules - 2

Product Rule: When multiplying like bases, add the exponents: $a^m \cdot a^n = a^{m+n}$ .
- Example 1: $2m^2 \cdot 2m^3 = 4m^5$
- Example 3: $4r^{-3} \cdot 2r^2 = 8r^{-1} = \frac{8}{r}$

Power of a Power Rule: Multiply exponents when raising a power to another power: $(a^m)^n = a^{m \times n}$ .
- Example 11: $(x^2)^0 = 1$
Power of a Product Rule: Distribute the exponent to all factors within the parentheses: $(ab)^n = a^n b^n$ .
- Example 14: $(4a^3)^2 = 16a^6$
- Example 12: $(2x^2)^{-4} = \frac{1}{(2x^2)^4} = \frac{1}{16x^8}$
Zero Exponent Rule: Any non-zero base raised to the power of zero is one: $a^0 = 1$ .
Negative Exponent Rule: To make an exponent positive, move the base to the denominator (or numerator): $a^{-n} = \frac{1}{a^n}$ .

Quotient Rule: When dividing like bases, subtract the exponent of the denominator from the exponent of the numerator: $\frac{a^m}{a^n} = a^{m-n}$ .
- Example 21: $\frac{r^2}{2r^3} = \frac{1}{2r}$
- Example 25: $\frac{3m^{-4}}{m^3} = 3m^{-7} = \frac{3}{m^7}$

Expressions are simplified by applying rules to each variable independently and handling numerical coefficients as standard fractions.

Simplify the following expression using the product rule: $5a^33a^4$ .
Apply the power of a power rule to find the value of: $(x^3)^2$ .
Use the power of a product rule to simplify: $(2x^2y)^3$ .
Calculate: $rac{6m^5}{2m^2}$ using the quotient rule.
Simplify using the negative exponent rule: ${3}{x^{-3}}$ .
Using multivariable simplification, simplify the expression: ${4ab^{-2}}{2a^2b^3}$ .
Evaluate: $(3x^2y^3)^0$ .