Properties_of_Exponents - Fundamentals -1

Vocabulary and mathematical properties based on the Kuta Software - Infinite Algebra 1 worksheet focusing on the manipulation and simplification of exponential expressions.

Product Rule of Exponents

The mathematical property applied in problems like $2m^2 \cdot 2m^3$ and $4a^3b^2 \cdot 3a^{-4}b^{-3}$, where you multiply the coefficients and add the exponents of common bases: $a^m \cdot a^n = a^{m+n}$.

Power of a Power Rule

The property used to simplify expressions like $(x^2)^0$ and $(4a^3)^2$, where you multiply the internal exponent by the external exponent: $(a^m)^n = a^{m \cdot n}$.

Power of a Product Rule

A rule seen in problems like $(4xy)^{-1}$ and $(2x^4y^{-3})^{-1}$, stating that a power applied to a product is equal to the product of each factor raised to that power: $(ab)^n = a^n b^n$.

Quotient Rule of Exponents

The property used to simplify fractions such as $\frac{r^2}{2r^3}$ and $\frac{3n^4}{3n^3}$, which involves subtracting the exponent in the denominator from the exponent in the numerator: $\frac{a^m}{a^n} = a^{m-n}$.

Zero Exponent Property

A rule applied in problems like $(x^2)^0$ and $4x^0$, where any non-zero base raised to the power of zero is defined as $1$, represented as $a^0 = 1$.

Negative Exponent Property

A rule used to simplify expressions like $m^{-3}$ or $(2b^4)^{-1}$ by moving the base to the opposite part of the fraction and making its exponent positive: $a^{-n} = \frac{1}{a^n}$.

Simplified Form (Positive Exponents)

The required state for final answers in these exercises, where each base appears only once and all exponents must be positive integers.

Coefficient

The numerical factor multiplied by variables in an expression, such as the $4$ and $2$ in $4n^4 \cdot 2n^{-3}$.

Multivariate Exponent Problems

Problems such as $\frac{2x^4y^{-4}z^{-3}}{3x^2y^{-3}z^4}$ that require applying exponent properties to multiple different variables ($x$, $y$, and $z$) simultaneously.