Achievable logoAchievable logo
GRE
Sign in
Sign up
Purchase
Textbook
Practice exams
Feedback
Community
How it works
Resources
Exam catalog
Mountain with a flag at the peak
Textbook
1. Welcome
2. Vocabulary approach
3. Quantitative reasoning
3.1 Quant intro
3.2 Arithmetic & algebra
3.2.1 Positive negative problems
3.2.2 Defined & undefined
3.2.3 GRE vocabulary list 01 (alacrity)
3.2.4 Odd even problems
3.2.5 GRE vocabulary list 02 (adulterate)
3.2.6 Algebra
3.2.7 Fraction math
3.2.8 GRE vocabulary list 03 (abstain)
3.2.9 Percent change
3.2.10 GRE vocabulary list 04 (anachronism)
3.2.11 Function problems
3.2.12 GRE vocabulary list 05 (ameliorate)
3.2.13 Divisors, prime factors, multiples
3.2.14 Greatest common factor (GCF) & Least common multiple (LCM)
3.2.15 GRE vocabulary list 06 (acumen)
3.2.16 Permutations and combinations
3.2.17 GRE vocabulary list 07 (aesthetic)
3.2.18 Decimals
3.2.19 GRE vocabulary list 08 (aggrandize)
3.2.20 FOIL and quadratic equations
3.2.21 GRE vocabulary list 09 (anodyne)
3.2.22 Exponent rules
3.2.23 GRE vocabulary list 10 (aberrant)
3.2.24 Square roots and radicals
3.2.25 Sequences
3.2.26 Venn diagrams & tables
3.2.27 Ratios
3.2.28 Mixtures
3.2.29 Probability
3.2.30 Algebra word problems
3.2.31 Number line, absolute value, inequalities
3.2.32 Simple and compound interest
3.2.33 System of linear equations (SOLE)
3.3 Statistics and data interpretation
3.4 Geometry
3.5 Strategies
4. Verbal reasoning
5. Analytical writing
6. Wrapping up
Achievable logoAchievable logo
3.2.22 Exponent rules
Achievable GRE
3. Quantitative reasoning
3.2. Arithmetic & algebra

Exponent rules

7 min read
Font
Discuss
Share
Feedback

Exponents can make a simple problem seem hard. However, with a clear understanding of the exponent rules, these problems can be much simpler than they look.

Here are the exponent rules that you will need to know.

Rule 1: With multiplication, add the exponents

34∗35=39

In this problem, both bases of 34 and 35 are 3. When you expand both exponents you get the following:

34×35=(3×3×3×3)×(3×3×3×3×3)=39

Notice how both exponents expand into many threes written in a row. This problem is just 3 multiplied by itself 9 separate times. The exponents, being 4 and 5, have a sum of 4+5=9.

Note that you can only add exponents if the base numbers are the same!

Rule 2: With division, subtract the exponents

36/33=33

Like before, the bases of 36 and 33 are 3. Thus, we can subtract the exponents. The subtraction is essentially the cancellation of three of the 3s on the top. The exponents, being 6 and 3, have a difference of 6−3=3.

36/33=(3×3×3×3×3×3)/(3×3×3)=3×3×3=33

Try this GRE problem that involves Rule 1 and 2:

0<x<1

Quantity A: 24(23)x3/25
Quantity B: 44x5/(4x)3

Do you know the answer?

(spoiler)

Answer: Quantity B is greater

Let’s simplify Quantity A first:

24(23)x3/2527x3/2522x34x3​

And let’s simplify Quantity B:

44x5/(4x)344x5/((43)(x3))44x2/4341x24x2​

Now we can compare them side by side:

A:B:​4x34x2​

Given that the question states 0<x<1, Quantity B must be greater. A fraction multiplied by itself shrinks, and a fraction multiplied by itself more will shrink more.

Rule 3: A negative exponent means an exponent on the other side of a fraction

3−4=1/(34)

Simply flip the fraction and rewrite the exponent without the negative in front. Notice how a negative exponent does not describe a negative number. Remembering Rule 2, we can see that 34/38 would be equal to 3−4 because 4−8=−4. Because four of the 3s on the bottom are canceled out by the four of the 3s on the top, we are left with 1/(34).

Rule 4: Any number to the power of 0 is 1

30=1

Remembering Rule 2, any exponented number minus that same number is 0. When we subtract equal exponents, we have an exponent of 0. Looking at it as a fraction, we can also see that everything on the top is equivalent to everything on the bottom. When the numerator equals the denominator, the fraction is always equal to 1.

33/33=33−3=30=1

Try this GRE question that involves Rule 3 and 4:

Which of the following represents y in terms of z?

Given: z−4z4x−y=xyxz

A. z
B. z/2
C. −z/2
D. 2/z
E. −2/z

Try to solve it, and check your work!

(spoiler)

Answer: C. −z/2

Let’s simplify the expression step by step.

z−4z4x−yz−4+4x−yz0x−y(1)x−yx−y1/xy11/xzx−z−z−z/2​=xyxz=xyxz=xyxz=xyxz=xyxz=xyxz=x2yxz=x2y=x2y=2y=y​

The way above illustrates how to use negative exponents to manipulate equations. However, you could have also simplified it in other ways, like this:

z−4z4x−yz−4+4x−yz0x−yx−yx−y−y0−z−z/2​=xyxz=xyxz=xyxz=xyxz=xy+z=y+z=2y+z=2y=y​

No matter which approach you take, you’ll get the same answer!

Rule 5: When a number with an exponent is raised to another exponent, multiply the exponents

(33)2=36

This can be rewritten as (3×3×3)×(3×3×3). Because there are six 3s multiplied in a row, this is equivalent to 36. Incidentally, if you multiply the exponents, you will get the same answer for the exponent 3×2=6.

Be careful when reading these questions! Note that (33)2 is not the same as 3(32).

(33)2=36=7293(32)=39=19683

Rule 6: When bases are different with the same exponents, multiply the bases and keep the exponent

33× 43=123

This problem is essentially (3×3×3)×(4×4×4). If you combine the pairs of 3s and 4s, you will get three 12s, i.e.: 12×12×12=123.

Try this GRE question that involves Rule 5 and 6:

Which mixed number represents this fraction?

((22)2)232 112​

A. 325625​
B. 425665​
C. 4256155​
D. 4256235​
E. 525665​

Give it a try and check your work!

(spoiler)

Answer: B. 425665​

Let’s simplify! We’ll write the fraction horizontally so it’s easier to follow:

32 112/((22)2)2(3∗11)2/((22)2)2332/((22)2)21089/((22)2)21089/(22)2∗21089/(22)41089/22∗41089/281089/256≈ 4.25390625​

We’ve simplified the expression, now what? There’s a few ways we can take it from here. To start, our final value is 4+0.25390625, so only the choices with a 4 whole number could be correct, eliminating choices A and E. Looking at just the fractional part 4+0.253906254, we can see that it is less than 0.5, which eliminates choices C and D since their fractional parts are greater than 0.5. That leaves us with choice B as the correct answer.

If you want to double-check (and you should to help avoid mistakes), you can plug in the fractional parts into your calculator. You’ll see that 65/256=0.25390625, confirming we’ve made the correct choice.

Turning radicals into exponents

It can be useful to translate radicals into exponents and use your exponent rules to solve for the answer. Remember, a radical can be described as a fraction exponent by flipping the radical root and inner exponent:

25​=2251​=251/2

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action:

Sign up for free to take 5 quiz questions on this topic

All rights reserved ©2016 - 2025 Achievable, Inc.