Exponents and roots can feel like a foreign language if you haven't thought about them in a few years. The notation looks strange. The rules seem arbitrary. And the GMAT® has a way of combining them in questions that make you feel like you're missing something obvious.
If that's where you are, you're in the right place. Exponents and roots are closely related — they're two sides of the same coin. Once you understand what they represent and how the rules connect, most of the confusion falls away.
This guide covers exponent rules, negative and zero exponents, roots, fractional exponents, and the common traps that catch students on test day. Everything you need to handle exponent and root questions on the GMAT® Focus Edition.
Where Exponents and Roots Appear on the GMAT® Focus Edition
Exponents and roots show up in two places on the GMAT® Focus Edition.
The Quantitative section includes exponents and roots in Problem Solving questions. You might need to simplify an expression, solve an equation, or compare values.
The Data Insights section includes them in Data Sufficiency questions. These often ask whether a statement is sufficient to determine the value of an expression involving exponents or roots.
The approach is the same in both sections. The difference is that DS asks whether you have enough information, while PS asks you to produce the answer. For a full breakdown of DS format and process, see our complete guide to GMAT® Data Sufficiency.
What Is an Exponent?
An exponent is a shortcut for repeated multiplication.
Instead of writing 5 × 5 × 5 × 5, you write 5 with a small 4 above and to the right. That means "multiply by 5, four times." The result is 625.
Two terms matter:
- The base is what you're multiplying. In 5 to the 4th power, the base is 5.
- The exponent is how many times you multiply by the base. In 5 to the 4th power, the exponent is 4.
You'll sometimes hear this read as "five to the fourth power" or "five to the power of four." Same thing. "Power" is just another word for exponent.
Why do exponents exist? Think about compound interest. If you have $100 growing at 10% per year and you want to know the value after 20 years, you'd need to multiply by 1.1 twenty times in a row. Writing that out by hand would be painful. Instead, you write (1.1) to the 20th power. Much cleaner.
An exponent is a shorthand that saves you from writing the same multiplication over and over. It's not a new kind of math — just a notation trick.
The Three Core Exponent Rules
Exponents have been used in finance, engineering, and computer science for a long time. Over the years, people discovered some basic patterns that always hold true. The GMAT® tests these patterns heavily, so they're worth knowing cold.
Rule 1: Multiplying Same Base — Add Exponents
When you multiply exponents with the same base, you add the exponents.
Example: 5 to the 3rd times 5 to the 4th equals 5 to the 7th.
Why does this work? Write it out. 5 to the 3rd means 5 × 5 × 5 (three fives). 5 to the 4th means 5 × 5 × 5 × 5 (four fives). Multiply them together and you get seven fives total — 5 to the 7th.
More examples:
7 to the 3rd times 7 to the 6th equals 7 to the 9th (3 + 6 = 9)
3 to the 10th times 3 to the 13th times 3 to the 8th equals 3 to the 31st (10 + 13 + 8 = 31)
Writing it out helps you see why the rule works. But on the test, you can just add the exponents.
Rule 2: Dividing Same Base — Subtract Exponents
When you divide exponents with the same base, you subtract the exponents.
Example: 5 to the 5th divided by 5 to the 3rd equals 5 to the 2nd.
Why? Think about it as a fraction. On top you have five fives multiplied together. On the bottom you have three fives multiplied together. Cancel three fives from the top with three fives from the bottom, and you're left with two fives on top. That's 5 to the 2nd.
More examples:
9 to the 7th divided by 9 to the 3rd equals 9 to the 4th (7 - 3 = 4)
14 to the 3rd divided by 14 to the 2nd equals 14 to the 1st, which is just 14 (3 - 2 = 1, and anything to the first power is the base itself)
Same idea as the multiplication rule — it's about counting how many times you're multiplying by the base. Division cancels some of those multiplications out.
Rule 3: Power to a Power — Multiply Exponents
When you take an exponent to another exponent, you multiply the exponents.
Example: (6 to the 4th) raised to the 10th power equals 6 to the 40th.
Here's why: 6 to the 4th means multiply by 6 four times. Raising that whole thing to the 10th means do that ten times. So you're multiplying by 6 a total of 4 × 10 = 40 times.
More examples:
(2 to the 5th) raised to the 7th equals 2 to the 35th (5 × 7 = 35)
(8 to the 3rd) raised to the 5th equals 8 to the 15th (3 × 5 = 15)
This rule is sometimes called "power to a power." Same concept — multiply the exponents.
All three rules are worth putting on flashcards if you're not already fluent with them. As we discussed in our complete guide to GMAT® Number Properties, having foundational rules memorized frees up mental energy for reasoning.
Negative Exponents
A negative exponent means division by the base instead of multiplication.
That's the key idea. Positive exponents mean "multiply by the base this many times." Negative exponents mean "divide by the base this many times."
Example: 5 to the negative 3rd means divide by 5, three times. That's the same as 15 to the 3rd, which is 1125.
You'll often hear this rule stated as: a negative exponent means one divided by the positive version. Same thing. 5 to the negative 3rd equals 15 to the 3rd.
Here's why it works. Think about what happens when you subtract 1 from an exponent:
5 to the 3rd = 5 × 5 × 5
5 to the 2nd = 5 × 5 (you divided by 5 to get here)
5 to the 1st = 5 (divided by 5 again)
5 to the 0th = 1 (more on this in a moment)
5 to the negative 1st = 15 (dividing by 5 past zero puts you in fractions)
Each time you subtract 1 from the exponent, you divide by the base. So negative exponents naturally represent division.
You can combine negative exponents with the other rules:
7 to the 6th times 7 to the negative 4th equals 7 to the 2nd (add exponents: 6 + (-4) = 2)
4 to the negative 9th divided by 4 to the 3rd equals 4 to the negative 12th (subtract: -9 - 3 = -12)
The GMAT® may give you answer choices in different forms. You might see 4 to the negative 12th written as 14 to the 12th. You should be comfortable recognizing that these are the same thing.
Zero Exponent
Any non-zero number raised to the power of zero equals 1.
5 to the 0th = 1. 100 to the 0th = 1. (-7) to the 0th = 1. 12 to the 0th = 1.
Here's why. Start with 5 to the 2nd = 25. Divide both sides by 5: you get 5 to the 1st = 5. Divide by 5 again: you get 5 to the 0th = 1.
Each time you subtract 1 from the exponent, you divide by the base. Going from 5 to the 1st to 5 to the 0th means dividing 5 by 5 — which gives you 1.
Almost any number divided by itself is 1. That's why almost any number to the zero power is 1.
The one exception is 0 to the 0th, which is undefined. Division by zero isn't allowed, so 0 to the 0th doesn't have a value. You almost certainly won't see this on the GMAT® — just know that dividing by zero is off limits.
This rule comes up on the GMAT® more than you might expect. As we cover in our guide to GMAT® timing strategy, knowing quick rules like this saves precious seconds on test day.
The Trap: Addition and Subtraction
This is where a lot of students get tripped up.
The exponent rules you just learned — adding exponents when multiplying, subtracting when dividing, multiplying for power to a power — those rules work for multiplication and division only.
They do NOT work for addition and subtraction.
Example:
2 to the 2nd times 2 to the 3rd = 2 to the 5th. That's correct — multiplication rule works.
2 to the 2nd plus 2 to the 3rd = 4 + 8 = 12. But 2 to the 5th is 32. Adding the exponents would give you the wrong answer.
This is one of the most common mistakes on the GMAT®. Under time pressure, it's tempting to see 2 to the 2nd plus 2 to the 3rd and think "just add the exponents." But that gives you 2 to the 5th = 32, which is way off from the correct answer of 12.
So what do you do when you see addition or subtraction with exponents?
You factor out a common term.
Example: 5 to the 7th plus 5 to the 8th
Is there a number that divides both 5 to the 7th and 5 to the 8th? Yes — 5 to the 7th divides both of them.
5 to the 7th divided by 5 to the 7th is 1.
5 to the 8th divided by 5 to the 7th is 5.
So you can rewrite: 5 to the 7th plus 5 to the 8th = 5 to the 7th times (1 + 5) = 5 to the 7th times 6.
Why is this useful? Because now you've turned addition into multiplication. And with multiplication, you can use all the exponent rules you know.
Train yourself to recognize this pattern. When you see addition or subtraction with exponents on the GMAT®, don't reach for the exponent rules. Factor out a common term instead. This single habit will prevent one of the most common exponent mistakes on the test.
For more on how to build this kind of systematic approach, see our guide to building GMAT® quant confidence.
Squaring and Cubing
The GMAT® sometimes uses specific terminology for common exponents. If you haven't thought about this in a while — especially if English isn't your first language — these terms are worth knowing.
When you raise a number to the power of 2, it's called "squaring" the number. "Five squared" means 5 to the 2nd, which is 25. The term comes from geometry: a square has four equal sides, and finding the area means multiplying the side length by itself.
When you raise a number to the power of 3, it's called "cubing" the number. "Four cubed" means 4 to the 3rd, which is 64. This also comes from geometry: a cube is a 3D box with equal sides, and finding the volume means multiplying the side length by itself three times.
You might also hear "the square of a number" (the result of squaring) or "the cube of a number" (the result of cubing). The square of 5 is 25. The cube of 4 is 64.
These terms show up on the GMAT®. If you're not familiar with them, put them on a flashcard.
What Is a Root?
A root is the inverse of an exponent.
If exponents build a number up by multiplying it by itself, roots work in the opposite direction. They figure out what number got multiplied to get there.
Example: 2 squared is 4. The square root of 4 asks the opposite question — what number, if you multiplied it by itself, would give you 4? The answer is 2.
So squaring and square roots are inverse operations. One builds up, the other works in reverse.
Square Roots
The square root is the most common type of root on the GMAT®.
When you see "the square root of 9," it's asking: what number can I multiply by itself to create 9? The answer is 3, because 3 × 3 = 9.
A few more:
The square root of 16 is 4 (because 4 × 4 = 16)
The square root of 25 is 5 (because 5 × 5 = 25)
The square root of 36 is 6 (because 6 × 6 = 36)
The Benchmarking Technique
You don't have a calculator on the GMAT® Quant section. (The Data Insights section has one, but with limited functionality.)
So what do you do when you don't know the square root of a number?
You benchmark.
Start with what you know and work your way up. Let's say you need the square root of 49:
2 × 2 = 4 (too small)
3 × 3 = 9 (too small)
4 × 4 = 16 (too small)
5 × 5 = 25 (too small)
6 × 6 = 36 (getting closer)
7 × 7 = 49. The answer is 7.
This might seem slow, but it works. And on the GMAT®, you'll usually be working with numbers that have clean roots — you just need to find them.
Perfect Squares to Memorize
Benchmarking works, but it's a lot faster if you already know the answer. That's why it's worth memorizing the perfect squares from 1 to 20.
A perfect square is a number you get by multiplying an integer by itself.
1 squared = 1
2 squared = 4
3 squared = 9
4 squared = 16
5 squared = 25
6 squared = 36
7 squared = 49
8 squared = 64
9 squared = 81
10 squared = 100
11 squared = 121
12 squared = 144
13 squared = 169
14 squared = 196
15 squared = 225
16 squared = 256
17 squared = 289
18 squared = 324
19 squared = 361
20 squared = 400
It's pretty unusual to see perfect squares larger than 20 squared on the GMAT®. It could happen, but you'd be prepared for most questions if you memorize these.
Cube Roots
The next most common root is the cube root.
The cube root asks: what number can I multiply by itself three times to produce this value?
The cube root of 27 is 3, because 3 × 3 × 3 = 27.
You can benchmark this too:
1 × 1 × 1 = 1 (too small)
2 × 2 × 2 = 8 (too small)
3 × 3 × 3 = 27. The answer is 3.
Perfect Cubes to Memorize
There are only four perfect cubes worth memorizing:
2 cubed = 8
3 cubed = 27
4 cubed = 64
5 cubed = 125
After that, it's so rare to see other cubes on the GMAT® that you probably don't need to worry about them.
Other Roots
The square root and cube root are the most common, but the concept works for any number. The fourth root of 16 is 2, because 2 × 2 × 2 × 2 = 16. The same logic extends to fifth roots, sixth roots, and so on.
How Roots Are Written
There are two main ways you'll see roots on the GMAT®.
The Radical Symbol
The first uses the radical symbol — √. It looks kind of like a check mark. The number under the bar is what you're taking the root of, and the bar extends over everything it covers.
For cube roots and other roots, a small number sits above the left side of the radical symbol. That number is the index — it tells you which root you're taking. A 3 means cube root, a 4 means fourth root. When there's no index, it's always a square root.
Fractional Exponents
The second way to write roots uses fractional exponents. This is where roots and exponents connect.
The key idea: a fractional exponent is just another way to write a root.
5 to the one-half power is the same as the square root of 5
5 to the one-third power is the same as the cube root of 5
5 to the one-fourth power is the same as the fourth root of 5
In general: x to the 1n power is the same as the nth root of x.
Why Fractional Exponents Equal Roots
Remember the first rule of exponents: when you multiply by the same base, you add the exponents.
Let's use that to figure out what 5 to the one-half means.
The square root of 5 asks: what number can I multiply by itself to get 5?
In exponent language: what number, when I multiply it by itself (add the exponents), gives me 5 to the 1st?
If you multiply 5 to the one-half times 5 to the one-half, you add the exponents: 12 + 12 = 1.
So 5 to the one-half times 5 to the one-half equals 5 to the 1st, which is 5. That's exactly what a square root does. So 5 to the one-half and the square root of 5 represent the same number.
The same logic works for cube roots. 8 to the one-third times 8 to the one-third times 8 to the one-third means adding the exponents: 13 + 13 + 13 = 1. That gives you 8 to the 1st, which is 8. So 8 to the one-third is the same as the cube root of 8, which is 2.
The Denominator-Numerator Rule
This is one of the most useful things to remember about fractional exponents:
The denominator of a fractional exponent tells you the root. The numerator tells you the exponent.
Let's say you see 27 to the two-thirds on the GMAT®.
The denominator is 3 — that means cube root.
The numerator is 2 — that means square it.
So you can break 27 to the two-thirds into two steps:
Step 1: Take the cube root of 27, which is 3
Step 2: Square that, which gives you 9
You can also do it the other way — square 27 first, then take the cube root. Both give you 9. But here's a useful tip: do the root first.
Why? Because taking the root usually makes the number smaller, which makes the next step easier. If you square 27 first, you get 729 — a much bigger number to work with by hand. If you take the cube root first, you get 3 — a much smaller number to square.
Train yourself to do the root first every time. The first time you try squaring before taking the root, you'll end up with a number like 729 and no calculator. Once you build the habit of root-first, you probably won't go back.
All Exponent Rules Apply to Roots
Here's something that might surprise you: all the rules of exponents also apply to roots.
Why? Because roots can be written as fractional exponents. And fractional exponents follow the same rules as regular exponents.
When multiplying roots with the same base, add the exponents.
When dividing roots with the same base, subtract the exponents.
When raising a root to a power, multiply the exponents.
Example: What is 4 to the one-half times 4 to the three-fourths?
Same base, so add the exponents: 12 + 34.
To add fractions, find a common denominator. 12 = 24. So 24 + 34 = 54.
The answer is 4 to the five-fourths.
Example: What is (4 to the one-half) cubed?
Power to a power, so multiply the exponents: 12 × 3 = 32.
The answer is 4 to the three-halves. Using the denominator-numerator rule: the square root of 4 is 2, and 2 cubed is 8.
Adding and Subtracting Roots
When you're adding or subtracting roots, the exponent rules don't directly apply. This is just like with regular exponents.
Instead, look for common factors you can pull out.
Example:
Notice that appears in both terms. You can factor it out:
This is the same factoring technique you use in basic algebra. The only difference is that the common factor happens to be a root.
If there's no common factor — like — there's no clean way to simplify. You'd just estimate if the GMAT® asks for a number.
Common GMAT® Exponent Traps
The GMAT® designs wrong answer choices to catch specific mistakes. Here are the ones to watch for.
Trap 1: Using Exponent Rules on Addition
This is the big one. You see 2 to the 3rd plus 2 to the 3rd and your brain wants to add the exponents to get 2 to the 6th. But 2 to the 3rd plus 2 to the 3rd is 8 + 8 = 16, which is 2 to the 4th, not 2 to the 6th.
The correct approach: 2 to the 3rd plus 2 to the 3rd = 2 times 2 to the 3rd = 2 to the 1st times 2 to the 3rd = 2 to the 4th.
Wrong answer choices on the GMAT® are designed to catch the "add the exponents" mistake. When you see addition with exponents, stop and factor.
Trap 2: Confusing Negative Exponents with Negative Bases
5 to the negative 4th isn't the same as (-5) to the 4th.
5 to the negative 4th = 15 to the 4th = 1625
(-5) to the 4th = 625
These look similar but mean very different things. The negative is in the exponent, not the base. Read carefully.
Trap 3: Forgetting the Root-First Rule
When you see 64 to the five-thirds, the temptation is to raise 64 to the 5th power first. That gives you an enormous number that's nearly impossible to take the cube root of by hand.
Instead, take the cube root first: the cube root of 64 is 4. Then raise 4 to the 5th: 4 × 4 = 16, 16 × 4 = 64, 64 × 4 = 256, 256 × 4 = 1,024.
Much more manageable. Root first, exponent second.
Trap 4: Mixing Up Squaring and Doubling
"Squared" means raised to the power of 2. "Doubled" means multiplied by 2. These are different operations.
5 squared = 5 × 5 = 25
5 doubled = 5 × 2 = 10
The GMAT® knows that students sometimes confuse these terms under pressure. Read carefully when you see "squared" or "doubled" in a question.
A Worked Example
Let's put the rules together with a GMAT®-style problem.
Question: Which of the following is equivalent to 3 to the 4th times 3 to the 5th3 to the 6th?
(A) 3 to the 3rd (B) 3 to the 5th (C) 3 to the 7th (D) 3 to the 15th (E) 3 to the 20th
Step 1: Numerator. 3 to the 4th times 3 to the 5th — same base, multiplication, so add exponents. 4 + 5 = 9. The numerator is 3 to the 9th.
Step 2: Divide. 3 to the 9th divided by 3 to the 6th — same base, division, so subtract exponents. 9 - 6 = 3.
The answer is 3 to the 3rd, which is (A).
The trap here is answer choice (D), 3 to the 15th. That's what you'd get if you added all the exponents together (4 + 5 + 6 = 15) without paying attention to which part is multiplication and which is division.
One rule at a time. That's how you avoid mixing up which rule to use when.
How to Study Exponents and Roots
If you want to build fluency with these rules, here's a simple approach.
First, make flashcards for the core rules. Multiplication means add. Division means subtract. Power to a power means multiply. Negative means divide. Zero means one. Addition and subtraction means factor.
Second, practice converting between radical form and fractional exponent form. The GMAT® may give you an answer in either form, and you need to recognize them as equivalent.
Third, drill the perfect squares (1 through 20) and perfect cubes (2 through 5) until they're automatic. This saves time on test day.
Fourth, when you review practice questions, pay special attention to exponent addition traps. If you got a question wrong because you added exponents when you should have factored, that's a sign you need to build the factoring habit. Our complete guide to GMAT® error tracking has a system for reviewing mistakes without the traditional error log overhead.
For more on how to structure your study plan around topics like this, see our complete guide to studying for the GMAT®.
FAQ
Do exponent rules work with addition and subtraction?
No. The core exponent rules — adding exponents for multiplication, subtracting for division, multiplying for power to a power — only work with multiplication and division. When you see addition or subtraction with exponents, factor out a common term instead. This is one of the most common exponent mistakes on the GMAT®.
What is a negative exponent?
A negative exponent means division by the base. 5 to the negative 3rd means divide by 5, three times, which equals 15 to the 3rd or 1125. You can also think of it as one divided by the positive version of the exponent.
What is a fractional exponent?
A fractional exponent is another way to write a root. The denominator tells you the root, and the numerator tells you the exponent. For example, x to the one-half power is the same as the square root of x. X to the one-third power is the same as the cube root of x.
How many perfect squares should I memorize for the GMAT®?
Memorize the perfect squares from 1 to 20. That covers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400. It's unusual to see perfect squares larger than 20 squared on the GMAT®.
Should I take the root first or the exponent first?
Take the root first. When a fractional exponent has a numerator greater than 1 — like 64 to the five-thirds — doing the root first (cube root of 64 = 4) makes the number smaller and the next step easier. Doing the exponent first (64 to the 5th = a very large number) makes the problem much harder to solve by hand.
Are exponents and roots the same thing?
They're closely related. A root is the inverse of an exponent — it undoes what an exponent does. Roots can also be written as fractional exponents, which means all the exponent rules apply to roots as well. The square root of x is the same as x to the one-half power. The cube root of x is the same as x to the one-third power.
What does "squaring" a number mean?
Squaring a number means raising it to the power of 2. "Five squared" = 5 to the 2nd = 25. The term comes from geometry — finding the area of a square means multiplying the side length by itself, which is the same as raising it to the second power.
What does "cubing" a number mean?
Cubing a number means raising it to the power of 3. "Four cubed" = 4 to the 3rd = 64. This also comes from geometry — finding the volume of a cube means multiplying the side length by itself three times.
Want to learn even more?
This guide covers the rules and the traps. But hearing someone talk through these concepts — explaining why the rules work, showing examples in real time — can make a big difference.
We cover exponents in detail in Lesson 16 of our GMAT® Math Basics series, available on Spotify, Apple Podcasts, and YouTube. Lesson 17 covers roots, fractional exponents, and how the exponent rules apply to roots.
If you're looking for a complete study system that ties all these topics together — from number properties to algebra to data sufficiency — start with our complete guide to studying for the GMAT®. It walks through how to build a study plan, when to take practice tests, and how to review your work.
For the full list of topics covered on the GMAT® Focus Edition, see our complete GMAT® topic list.
And if you're feeling stuck on quant more broadly, our guide to building GMAT® quant confidence walks through a framework for getting from "I'm bad at math" to handling GMAT® quant questions with confidence.