- 🔢 Number systems: This includes real, irrational, and complex numbers. You’ll be asked to perform arithmetic operations with these numbers, such as addition, multiplication, etc.
- 🧮: Rational exponents: These are exponents written in the form x^p/q, where q is not equal to 0. Similar to number systems, you’ll also be asked to perform arithmetic operations with these numbers, in which you must know and apply the properties of rational exponents to answer these questions correctly.
- 🖥️ Vectors/matrices: These tend to show up the least on the exam, but these are still important topics to know for the test. You’ll be asked to perform various operations with numbers in the matrices and vectors. You may also be asked how to mathematically write an equation for a vector based on a graph/description given.
How to Tackle Numbers and Quantity Questions
🔍 Understanding the properties of each type of number/mathematical expression
Since this section focuses on different types of numbers, such as real, imaginary, exponents, and matrices, it is important that you know how these mathematical expressions and numbers behave 🔢 . Unlike algebra, where you could plug in a number to guide you to your answer, many of these questions will require prior knowledge of the subject (ie. imaginary numbers, radical exponents, etc.) 👩🏫, so make sure you solidify your understanding of these concepts prior to test day. Revisiting and practicing these concepts can make it easier for you to recognize and solve similar problems on test day. ✅
🧠 Mental math is your best friend!
Cut corners in timing using mental math! ⏳ Once you get through most of a problem, you’ll likely end up with something that can easily be solved using mental math. Take advantage of this and do not pull out your calculator 💻 as this will waste precious time that can be used to tackle and solve more difficult questions. Timing is key to doing well on this test, so make sure you’re using every second efficiently. Be careful with this though, as mental math can also lead to simple, yet costly mistakes.
🤓 Identify Patterns and Relationships
Many questions in this category involve patterns and relationships between numbers. Train yourself to recognize these patterns 🏃♂️, as it can lead you to the correct answer more quickly, saving time with also being efficient.
➡️ How to Approach a Real/Complex Number Question
To answer this question, remember how the imaginary number i works. The formula is already given here, so if you were to plug in √ -1 into the equation, you would get (√ -1 - 1 - √-1)/(-√-1 + 1 + √-1). Now, you can subtract √-1 - √-1 from the numerator and - √-1 + √-1 from the denominator to bring the equation to -1/1, which equals to 1.
This process is lengthy, however, so it is better if you understand the powers of i. i^1 is i and i^2 is -1, as multiplying two radical expressions of the same base will cancel out the radical, leaving the answer to be a real number. At i^3, it is the same as multiplying -1 with i (also can be written as i^2 * i), which is -i. At i^4, this is equivalent to multiplying √-1 * √-1 (or i^2 * i^2), which equals to 1. Once you get to i^5 though, the answer is i again. Based on this, you can see a pattern forming, repeating after every 4 powers of i. Knowing this pattern is important when encountering similar ACT questions. Here is a table of these powers for better understanding:
| i¹ | √-1 |
| i² | √-1 * √-1 = -1 |
| i³ | √-1 * √-1 * √-1 = -1 * √-1 = - √-1 = -i |
| i⁴ | √-1 * √-1 * √-1 * √-1 = -1 * -1 = 1 |
| i⁵ | √-1 * √-1 * √-1 * √-1 * √-1 = -1 * -1 * √-1 = 1 * √-1 = √-1 = i |
➡️ How to Approach a Rational Exponent Question
This is one type of question you might found on the ACT math section that involves rational exponents. Here, it is important to understand the manipulate and rewrite exponents to make them easier to solve. Also, finding the fastest way to solve this problem will save you time to use on more difficult and lengthy questions.
Firstly, to remove a negative exponent, you must flip the fraction, making it (64/27)^⅔. Since 64 and 27 are cubic products, you can rewrite them as 4^3 and 3^3 respectively, making the expression (4^3 / 3^3) ^⅔. Now, you can cancel the exponents, since ⅔ has the denominator of 3 and the exponents inside of the parentheses are also 3s, making the expression (4/3)^2. Square the fraction to get 16/9, which is J.
➡️ How to Approach a Vector Question
Here, we see that the parameters for the vector are given: i is east/west or right/left and j is north/south or up/down. Going to the east (right) or north (up) indicates a positive (+) movement along the graph, while going to the west (left) or south (down) indicates a negative (-) movement along the graph. With that being said, we can translate the following text into an equation. Since Maria is going 12 miles south, we will be using the vector j, and this would be written as -12j mathematically.
Sometimes, you would questions like this but in a graph, asking you to derive the equation based on the vector shown on the graph. It follow a similar pattern of solving, making these questions pretty easy to solve once you understand how vectors are drawn and expressed.
➡️ How to Approach a Matrices Question
For this question, you’ll need to understand how matrices are multiplied. This is different from scalar multiplication, which involves multiplying a matrix by a whole number. In multiplying one matrix with another, the number in the first row and column in matrix X, which is -1, will be multiplied by the number in the first row and column of matrix Y, which is -2, making the product of these numbers 2. After, the number in the first row and second column of matrix X, which is 0, will be multiplied by the number in the second row and first column of matrix Y, which is -1, making the product 0. Omitting the 0, you are left with 2 as your final answer, which is D.
Here, it is important to know matrix multiplication, which can get more complicated with more rows and columns, though this does not show up on the ACT often.
😮💨 Conclusion
While some of these questions may look a little daunting in the beginning, with a little manipulation and rewriting of expressions, these questions are targets for easy points in the exam. In addition, knowing the properties and rules that come with performing arithmetic operations with various types of numbers that appear on the ACT math section will be helpful in recognizing them early and tackling them efficiently! 🙌
