College Algebra: Polynomials, Exponent Rules, Factoring, Rational Expressions, and Graphs

Exponential Notation Fundamentals
- In the expression $a^n$ , $a$ represents the base and $n$ represents the exponent (or power).
- Definition: For any positive integer $n$ , the expression $a^n$ denotes the repeated multiplication of base $a$ by itself $n$ times: $a^n = a \cdot a \cdot a \cdots a \quad (n\text{ factors of } a)$
- Verbal Expressions and Terminology:
- $a^n$ is spoken as " $a$ to the $n\text{th}$ power".
- $a^2$ is specifically termed " $a$ squared".
- $a^3$ is specifically termed " $a$ cubed".
- Structural Diagram:
- Expanded Factor Meaning Examples:
- $4^3 = 4 \cdot 4 \cdot 4 = 64$
- $(87.2)^6 = (87.2)(87.2)(87.2)(87.2)(87.2)(87.2)$
- $(-1.1)^5 = (-1.1)(-1.1)(-1.1)(-1.1)(-1.1)$
- $(\frac{2}{3})^4 = (\frac{2}{3})(\frac{2}{3})(\frac{2}{3})(\frac{2}{3}) = \frac{16}{81}$
- $(11c)^3 = (11c)(11c)(11c) = (11)(11)(11) \cdot c \cdot c \cdot c = 1331c^3$
- $-6y^4 = -6 \cdot y \cdot y \cdot y \cdot y$ (the base is solely $y$ , not $-6y$ ; the coefficient $-6$ is applied once)
- $(x + 5)^3 = (x + 5)(x + 5)(x + 5)$
Rules for Exponents of 0 and 1
- Exponent of 1: For any real number $a$ : $a^1 = a$
- Exponent of 0: For any nonzero real number $a$ ( $a \neq 0$ ): $a^0 = 1$
- Caveat: The expression $0^0$ is indeterminate and undefined in standard arithmetic.
- Evaluation Examples: *