These are all the rules for exponents on the ACT® Math. You might see the table below and think "woahhhh, how am I ever supposed to remember this?" There's an ACT® "hack" that makes these questions, for the most part, easier than they may look. Even if you memorized the table below from pre-calc, you can still save time with this trick.
Exponent Rule Table
| Rule | Expression |
|---|---|
| a^(m) aⁿ | a^(m+n) |
| a^(m)/aⁿ | a^(m-n) |
| (ab)^(m) | a^(m) b^(m) |
| (a^(m))ⁿ | a^(mn) |
| a^(-m) | 1/a^(m) |
| a⁰ | 1 |
| ^(a)√mⁿ | m^(n/a) |
Radical Expressions
Radicals are another way to express roots or fractional exponents. Here's a quick overview:
- √x = x^(1/2)
- ∛x = x^(1/3)
- √x * √x = x
To be clear, if you know (a^(m))ⁿ = a^(mn), and if that makes sense to you, then you can solve the question that way. More advanced questions sometimes benefit from the approach below.
An Easy Way to Solve These Sorts of Questions
A strategy I use for number substitution on the ACT® Math is to acknowledge that the variables we're given in equations don't actually represent real numbers.
In simpler terms, you can't go to the store and buy a oranges and b apples. You can buy 2 oranges and 3 apples, which are real numbers the variables a and b could represent.
It's so much more practical to multiply 2 * 3 than it is a * b. I use real numbers when they give me variables on the ACT® Math. We'll see more practice on this below. Suffice it to say for now, if I wanted to know what (a^(m))ⁿ is equal to, I could assign real numbers to a, m, and n.
Consider This Example
Let's work through an example to see how this strategy works.
Question:
Which of the following is equivalent to (a^(m))ⁿ?
- A) a^(mn)
- B) a^(m+n)
- C) a^(-m)
- D) a^(m^n)
- E) a^(m-n)
Click for the Answer
Correct Answer: A) a^(mn)
Explanation:
We can verify this by substituting real numbers.
Let's choose a = 2, m = 3, and n = 4.
(2³)⁴ = 8⁴ = 4096
Now, let's check option A: 2 ^(3*4) = 2¹² = 4096
This confirms that (a^(m))ⁿ = a^(mn)
More Practice
Let's apply our number substitution strategy to more complex problems.
Example 1: Simplifying Fractional Exponents
Which of the following is equal to a^(m)/aⁿ?
- A) a^(m+n)
- B) a^(m-n)
- C) a^(m⋅n)
- D) a^(m^n)
- E) a^(n-m)
Click for the Answer
Correct Answer: B) a^(m-n)
Explanation:
Let's use a = 3, m = 5, and n = 2. 3⁵/3
² = 243/9 = 27 Now, let's check option B: 3
⁵⁻² = 3³ = 27 This confirms that a
^(m)/aⁿ = a^(m-n)
Example 2: Understanding Negative Exponents
Which of the following is equal to a^(-m)?
- A) 1/a^(m)
- B) a^(m)
- C) m^(a)
- D)
- E) a^(m-1)
Click for the Answer
Correct Answer: A) 1/a^(m)
Explanation:
Let's use a = 5 and m = 3. 5⁻³ = 1/125
Now, let's check option A: 1/5³ = 1/125
This confirms that a^(-m) = 1/a^(m)
Example 3: Working with Radicals and Exponents
Which of the following is equal to ^(b)√(m^(a))?
- A) m^(a/b)
- B) m^(b/a)
- C) m^(a⋅b)
- D) m^(a+b)
- E) m^(a-b)
Click for the Answer
Correct Answer: A) m^(a/b)
Explanation:
Let's use m = 9, a = 6, and b = 3.
³√(9⁶) = 81
Now, let's check option A:
9^(6/3) = 9² = 81
This confirms that ^(b)√(m^(a)) = m^(a/b)
Real-World Application of Exponents
Exponents show up in real-life situations more often than you might think. Take compound interest, for example. If you put $1000 in a savings account with 5% annual interest, your balance after t years is calculated using this formula:
1000 * (1.05)^(t)
That exponent is what makes compound interest grow your money faster than simple interest. It's a practical application of exponents that directly impacts your finances.