ACT Math Skill Review: Algebraic Manipulation

What’s with All the Letters?

Many of the questions on the ACT math test may not include numbers at all. These questions (and others) will test your knowledge of basic algebraic concepts. You will need to be comfortable working with algebraic expressions and functions, solving for an unknown variable, factoring and simplifying algebraic expressions, and applying algebraic concepts to problem solving.

Definitions and Basics

An algebraic expression is a collection of terms combined by addition, subtraction, or both in which letters or variables are used to represent numbers. Terms are made up of numbers or variables combined by multiplication or division. For SAT success, you will need to be able to apply the basic operations of arithmetic to algebraic expressions.

For example, like terms can be combined:

Algebraic Exponents

Let’s take a look at some definitions for algebraic exponents:

Here are some key concepts to remember:

• Add exponents when multiplying expressions with the same base:

  

• Subtract exponents when dividing expressions with the same base:

  

• Multiply exponents when a number raised to an exponent is raised to a second exponent:

  

Factoring

You are likely to see questions that ask you to evaluate or compare expressions that require factoring. Here is one key piece of advice:

Rarely will a math question put an expression in the most helpful format. So if it’s already factored, you might want to combine it. If it’s not factored, you’ll probably need to factor it.

The types of factoring included in the math section are:

• Difference of two squares

  

• Finding common factors or grouping common terms

  

• Factoring quadratics and polynomials

  

Examples

Answers and Explanations

1. The correct answer is B. This question requires you to follow the order of operations (PEMDAS), manipulate exponents and combine like terms. First you multiply the exponents to get:Next, you add the exponents to arrive at the answer:
    
2. The correct answer is H. You could multiply out all of the answer choices to find the one that matches the expression in the question. However, it’s faster to simply factor the expression. First, factor out the greatest common factor of each of the values (6) to get:Then, factor the algebraic expression within the parentheses to get: