Math 110 Test 1 Study Guide: Fundamental Concepts of Algebra and Equations

Test Format and Overview

This study guide outlines the structure and key topics for the Math 110 Test 1. The exam is closed book and closed notes, with no calculators allowed. Students are expected to show all work, and partial credit will be awarded where appropriate.

Topics Covered

• Rules of Exponents and Radicals

• Factoring

• Rational Expressions (expressions with variables in the denominator)

• Quadratic Equations and Equations in Quadratic Form

• Rational Equations (equations with variables in the denominator)

• Absolute Value Equations and Rational/Quadratic Expressions

• Solving Equations with Several Variables for One Variable

• Determining Mathematical Statements' Truth Values

Formulas You Should Know

• Exponent Rules:

  • $a^m \cdot a^n = a^{m+n}$

  • $\frac{a^m}{a^n} = a^{m-n}$

  • $(a^m)^n = a^{mn}$

  • $(ab)^n = a^n b^n$

  • $a^{-n} = \frac{1}{a^n}$

  • $a^{1/n} = \sqrt[n]{a}$

• Quadratic Formula:

  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Key Skills and Concepts

• Simplifying Expressions: Includes absolute value, exponents, radicals, and rational expressions.

• Factoring: Factoring polynomials and recognizing special products.

• Solving Equations: Linear, quadratic, rational, and absolute value equations.

• Solving for a Variable: Rearranging formulas to isolate a specified variable.

• Mathematical Reasoning: Determining the truth or falsity of mathematical statements.

Examples and Applications

• Exponent Example: Simplify $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$.

• Factoring Example: Factor $x^2 - 9 = (x - 3)(x + 3)$.

• Quadratic Equation Example: Solve $x^2 - 5x + 6 = 0$ using factoring: $(x-2)(x-3)=0 \implies x=2, 3$.

• Rational Expression Example: Simplify $\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x+2$ (for $x \neq 2$).

• Solving for a Variable Example: Given $A = lw$, solve for $w$: $w = \frac{A}{l}$.

Table: Exponent Rules Summary

Rule	Formula	Example
Product of Powers	$a^m \cdot a^n = a^{m+n}$	$2^3 \cdot 2^2 = 2^5 = 32$
Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	$5^4 / 5^2 = 5^{2} = 25$
Power of a Power	$(a^m)^n = a^{mn}$	$(3^2)^3 = 3^{6} = 729$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$4^{-2} = 1/16$
Fractional Exponent	$a^{1/n} = \sqrt[n]{a}$	$8^{1/3} = 2$

Additional info:

• Students should be comfortable with all forms of algebraic manipulation, including recognizing when to factor, how to simplify complex fractions, and how to apply the quadratic formula.

• Understanding the properties of exponents and radicals is foundational for success in later topics such as functions and graphing.