Properties_of_Exponents - Fundamentals -1

Page 1: Fundamental Exponent Properties

Multiplication and Power Rules

• Product Rule: For any base $a$, $a^m \cdot a^n = a^{m+n}$.

    - Example: $2m^2 \cdot 2m^3 = 4m^5$ (Prob. 1).

    - Example: $4a^3b^2 \cdot 3a^{-4}b^{-3} = 12a^{-1}b^{-1} = \frac{12}{ab}$ (Prob. 9).

• Power Rule: To raise a power to another power, multiply the exponents ($(a^m)^n = a^{m \times n}$).

    - Distribution: $(4a^3)^2 = 16a^6$ (Prob. 14).

    - Negative Powers: $(2x^2)^{-4} = \frac{1}{16x^8}$ (Prob. 12).

• Zero Exponent Property: Any non-zero base raised to the power of zero is one ($a^0 = 1$).

    - Example: $(x^2)^0 = 1$ (Prob. 11).

Page 2: Quotient Rules and Multi-Variable Simplification

Division and Quotient Properties

• Quotient Rule: To divide terms with the same base, subtract the exponent of the denominator from the numerator ($\frac{a^m}{a^n} = a^{m-n}$).

    - Example: $\frac{r^2}{2r^3} = \frac{1}{2r}$ (Prob. 21).

    - Example: $\frac{3m^{-4}}{m^3} = \frac{3}{m^7}$ (Prob. 25).

Complex Variable Reductions

The worksheet concludes with expressions involving multiple variables ($x$, $y$, $z$, $h$, $j$, $k$, $m$, $n$, $p$) and various coefficients.

• Multi-Variable Fraction Simplification:

    - Prob. 26: $\frac{2x^4y^{-4}z^{-3}}{3x^2y^{-3}z^4} = \frac{2x^2}{3yz^7}$

    - Prob. 29: $\frac{4m^4n^3p^3}{3m^2n^2p^4} = \frac{4m^2n}{3p}$

    - Prob. 30: $\frac{3x^3y^{-1}z^{-1}}{x^{-4}y^0z^0} = \frac{3x^7}{yz}$

Additional Practice Problems

1. Simplify: $2a^3 \cdot 4a^5$

2. Simplify: $\frac{15b^5}{3b^2}$

3. Simplify: $\left(5x^{-3}\right)^3$

4. Given: $\frac{m^4n^3}{m^2n^5}$, simplify to find $\frac{m^{?}}{n^{?}}$.

5. Simplify: $\frac{(x^2y^{-1})^2}{xy^3}$.