🎒ACT Review

This category covers about 4–6 questions on the ACT Math section. That's a small slice, but on a test where every point matters, these are often quick wins if you know the underlying rules. The topics here draw mostly from Algebra 2 and Precalculus, so if you've taken those courses, you're already in good shape.

What Topics to Expect

Number Systems (Real, Irrational, and Complex Numbers) You'll need to perform arithmetic with different types of numbers. The most commonly tested piece is working with the imaginary number $i = \sqrt{-1}$ . Questions typically ask you to simplify expressions involving $i$ or combine complex numbers.

Rational Exponents These are exponents written as fractions, like $x^{p/q}$ where $q \neq 0$ . The key idea: $x^{p/q}$ means "take the $q$ th root of $x$ , then raise it to the $p$ th power." You'll need to apply exponent rules (negative exponents, power of a power, etc.) to simplify expressions.

Vectors and Matrices These show up the least, but they still appear. Vector questions usually ask you to translate a description or graph into component form using $\mathbf{i}$ and $\mathbf{j}$ . Matrix questions typically involve multiplication or scalar operations.

How to Tackle Number and Quantity Questions

Know the properties before test day. Unlike algebra questions where you can sometimes plug in numbers and work backward, these questions rely on you already understanding how imaginary numbers, rational exponents, and matrices behave. There's no shortcut around learning the rules, so make sure you've practiced them before the exam.

Use mental math when you can. Once you've simplified an expression down to its last step or two, you can often finish in your head rather than reaching for your calculator. This saves real time across the section. Just be careful: mental math also invites careless errors, so double-check when the arithmetic gets tricky.

Look for patterns. Many of these questions reward pattern recognition. The powers of $i$ cycle every four. Rational exponents often simplify cleanly when you spot perfect squares or cubes. Training yourself to notice these shortcuts makes you both faster and more accurate.

How to Approach a Real/Complex Number Question

The most important thing to know for complex number questions is how the powers of $i$ cycle. Here's the pattern:

Power	Expansion	Result
$i^1$	$\sqrt{-1}$	$i$
$i^2$	$i \cdot i$	$-1$
$i^3$	$i^2 \cdot i = -1 \cdot i$	$-i$
$i^4$	$i^2 \cdot i^2 = (-1)(-1)$	$1$
$i^5$	$i^4 \cdot i = 1 \cdot i$	$i$

Notice that after $i^4$ , the cycle repeats: $i, -1, -i, 1, i, -1, \ldots$ This means for any power of $i$ , you just need to find the remainder when you divide the exponent by 4.

Example Walkthrough

Suppose a question gives you the expression $\frac{i - 1 - i}{-i + 1 + i}$ and asks you to simplify.

In the numerator, the $i$ and $-i$ cancel, leaving $-1$ .
In the denominator, the $-i$ and $i$ cancel, leaving $1$ .
The expression simplifies to $\frac{-1}{1} = -1$ .

The key was recognizing which terms cancel rather than trying to compute everything from scratch.

How to Approach a Rational Exponent Question

Rational exponent questions test whether you can manipulate exponent rules fluently. Here's a typical process.

Example: Simplify $\left(\frac{27}{64}\right)^{-2/3}$

Handle the negative exponent. A negative exponent means you flip the fraction: $\left(\frac{27}{64}\right)^{-2/3} = \left(\frac{64}{27}\right)^{2/3}$
Recognize perfect powers. Notice that $64 = 4^3$ and $27 = 3^3$ . Rewrite: $\left(\frac{4^3}{3^3}\right)^{2/3}$
Apply the power-of-a-power rule. Multiply the exponents: $3 \times \frac{2}{3} = 2$ . This gives: $\left(\frac{4}{3}\right)^{2}$
Square the fraction. $\frac{4^2}{3^2} = \frac{16}{9}$

The answer is $\frac{16}{9}$ .

The biggest time-saver here was spotting that 64 and 27 are perfect cubes. Train yourself to recognize common perfect squares (4, 9, 16, 25, 36...) and perfect cubes (8, 27, 64, 125...) so you can apply this technique quickly.

How to Approach a Vector Question

ACT vector questions usually give you a real-world description of movement and ask you to express it in vector notation.

The standard setup: $\mathbf{i}$ represents east/west (positive = east, negative = west) and $\mathbf{j}$ represents north/south (positive = north, negative = south).

Example: Maria travels 12 miles south.

South is the negative $\mathbf{j}$ direction.
The vector is $-12\mathbf{j}$ .

That's it. If she traveled 5 miles east and 12 miles south, the vector would be $5\mathbf{i} - 12\mathbf{j}$ .

You might also see graph-based versions where a vector is drawn on a coordinate plane and you need to write the equation. The approach is the same: read the horizontal component for $\mathbf{i}$ and the vertical component for $\mathbf{j}$ , paying attention to sign.

How to Approach a Matrices Question

Matrix questions on the ACT are usually straightforward, but you need to know the mechanics of matrix multiplication (it's not just multiplying corresponding entries).

How matrix multiplication works for a 1×2 times 2×1 case:

To find the entry in the result, you multiply across the row of the first matrix and down the column of the second matrix, then add the products.

Example: Multiply $\begin{bmatrix} -1 & 0 \end{bmatrix} \times \begin{bmatrix} -2 \\ -1 \end{bmatrix}$

Multiply the first entry of the row by the first entry of the column: $(-1)(-2) = 2$
Multiply the second entry of the row by the second entry of the column: $(0)(-1) = 0$
Add the products: $2 + 0 = 2$

The answer is $2$ .

For larger matrices, you repeat this row-by-column process for each entry in the result. The ACT rarely goes beyond 2×2 matrices, so mastering this basic pattern is usually enough.

Conclusion

These questions reward preparation more than raw problem-solving. If you memorize the cycle of powers of $i$ , practice simplifying rational exponents, and understand the basics of vector notation and matrix multiplication, you'll be able to move through these 4–6 questions quickly and bank time for harder problems elsewhere on the test.

🎒ACT Review