The Timpson Lyceum · College Algebra

Lecture 7 (College Algebra)

Overview

This lecture builds the algebraic vocabulary that the rest of the course relies on, moving from arithmetic of powers up through the language of sets. It defines integer exponents and proves the three core exponent laws (product, power-of-a-power, and quotient), then extends the definition to the zero exponent. From there it introduces sets, set membership, and set-builder notation, formalizes the idea of a variable versus a constant, classifies polynomials by their building-block monomials, and finishes with a careful treatment of square roots — including why a negative number has no real square root and why $\sqrt{x^2} = |x|$.

Key Concepts

Absolute value (review). For any real number $x$, $|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$ So if $a < 0$, then $|a| = -a$ is a positive number (e.g. $|-3| = -(-3) = 3$).

Definition of a positive-integer exponent. For a number $a$ and a natural number $m$, $a^m$ is the product of $m$ copies of $a$: $a^m = \underbrace{a \cdot a \cdots a}_{m \text{ times}}$ We name $a^2$ "$a$ squared" (the area of a square of side $a$) and $a^3$ "$a$ cubed" (the volume of a cube of side $a$).

Exponent laws (each provable by counting factors).

• Product rule: $a^m \cdot a^n = a^{m+n}$. Multiplying writes $m$ copies and then $n$ copies of $a$, for $m+n$ copies total.
• Power of a power: $(a^m)^n = a^{mn}$. Raising to the $n$th power lists $a^m$ a total of $n$ times; adding the exponents gives $\underbrace{m+m+\cdots+m}_{n} = mn$.
• Quotient rule (for $a \ne 0$): three cases.
  • $\dfrac{a^m}{a^m} = 1$ (a nonzero number over itself).
  • $\dfrac{a^m}{a^n} = a^{m-n}$ when $m > n$.
  • $\dfrac{a^m}{a^n} = \dfrac{1}{a^{\,n-m}}$ when $n > m$.

Zero exponent. For $a \ne 0$ we define $a^0 = 1$. This is consistent with the other rules: $a^n \cdot a^0 = a^{n+0} = a^n$, and $\dfrac{a^n}{a^n} = a^{n-n} = a^0 = 1$. ($a^0$ is the empty product, just as an empty sum is $0$.)

Sets. A set is any unordered collection of unique objects, written between curly braces. Because order and duplicates do not matter, $\{1,2,3\} = \{3,2,1\} = \{1,1,2,3\}$

• Membership: $x \in A$ means $x$ is an element of the set $A$.
• The empty set $\varnothing = \{\,\}$ contains nothing and plays a role for sets like $0$ does in arithmetic.
• Common number sets: naturals $\mathbb{N} = \{1,2,3,\dots\}$, integers $\mathbb{Z} = \{\dots,-1,0,1,\dots\}$, rationals $\mathbb{Q}$ (all fractions), reals $\mathbb{R}$.

Set-builder notation. Specify a dummy variable, a colon, and a condition: $\{\,x : P(x)\,\}$. If the variable is drawn from a known set, that set is usually placed before the colon. Examples: $\mathbb{N} = \{\, m \in \mathbb{Z} : m \ge 1 \,\}, \qquad \mathbb{Q} = \{\, x \in \mathbb{R} : \exists\, m,n \in \mathbb{Z},\ n \ne 0,\ x = \tfrac{m}{n} \,\}$

Variable vs. constant. A variable is a symbol (typically $x,y,z$) standing for any element of a non-empty set $S$, its domain; each member of $S$ is a possible value. Saying "$x < 2$" is the same as "$x \in S$" where $S = \{\,x \in \mathbb{R} : x < 2\,\}$. A constant (typically $a,b,c$) represents one fixed number. In $A = \pi r^2$, $\pi$ is a constant, $r$ is the independent variable, and $A$ is the dependent variable.

Monomials and polynomials.

• A monomial is a nonzero coefficient times one or more variables raised to non-negative integer exponents (e.g. $7ax^2y^3$). Its degree is the sum of the variable exponents — so $7ax^2y^3$ has degree $5$ (the constant $a$ contributes nothing). Any nonzero constant is a degree-$0$ monomial, since $9 = 9x^0$.
• A polynomial is a sum of monomials. Its degree is the largest degree among its terms. Special names: a binomial has two terms, a trinomial has three, and a degree-$1$ polynomial is linear (e.g. $bx + c$, the form of a line).

Square roots. $x$ is a square root of $b$ provided $x^2 = b$.

• Every positive $b$ has two square roots, written $+\sqrt{b}$ and $-\sqrt{b}$. The radical symbol $\sqrt{b}$ always denotes the non-negative root — e.g. $\sqrt{16} = 4$, and the other root is $-\sqrt{16} = -4$.
• $0$ has exactly one square root, $0$.
• A negative number has no real square root, because any real number squared is $\ge 0$.
• Identity: $\sqrt{x^2} = |x|$ for every real $x$.

Perfect squares. A rational number is a perfect square if it equals the square of another rational, e.g. $\dfrac{169}{25} = \left(\dfrac{13}{5}\right)^2$. The idea extends to monomials and to quotients of monomials.

Worked Example

Simplify $\dfrac{a^{10}}{a^{2}}$ using the quotient rule, then simplify $\left(\dfrac{3}{2}\right)^4$.

A short representative proof — why $\sqrt{x^2} = |x|$. Take any real $x$ and check the three cases.

In every case $\sqrt{x^2} = |x|$. For instance $\sqrt{(-5)^2} = |-5| = 5$.

Summary

• $a^m$ means $m$ copies of $a$ multiplied; the exponent laws $a^m a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, and $\dfrac{a^m}{a^n} = a^{m-n}$ all follow from counting factors.
• $a^0 = 1$ for $a \ne 0$ — the empty product — and it is consistent with every exponent rule.
• A set is an unordered collection of distinct elements; set-builder notation $\{\,x \in S : P(x)\,\}$ defines a set by a membership condition, with $\varnothing \subseteq \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ as the standard number systems.
• A variable ranges over its domain set while a constant is a single fixed value; convention uses $x,y,z$ for variables and $a,b,c$ for constants.
• A monomial is a coefficient times variables with non-negative integer exponents, with degree equal to the sum of those exponents; a polynomial is a sum of monomials whose degree is its highest-degree term (binomial, trinomial, and linear name special cases).
• The radical $\sqrt{b}$ always returns the non-negative root; positive numbers have two square roots, $0$ has one, negatives have none in $\mathbb{R}$, and $\sqrt{x^2} = |x|$.