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3.2.22 Exponent rules
Achievable GRE
3. Quantitative reasoning
3.2. Arithmetic & algebra

Exponent rules

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

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

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

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

Rule 2: With division, subtract the exponents

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

Try this GRE problem that involves Rule 1 and 2:

Quantity A:
Quantity B:

Do you know the answer?

(spoiler)

Answer: Quantity B is greater

Let’s simplify Quantity A first:

And let’s simplify Quantity B:

Now we can compare them side by side:

Given that the question states , 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

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 would be equal to because . Because four of the s on the bottom are canceled out by the four of the s on the top, we are left with .

Rule 4: Any number to a 0 exponent is 1

Remembering Rule 2, any exponented number minus that same number is . When we subtract equal exponents, we have an exponent of . 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 .

Try this GRE question that involves Rule 3 and 4:

Which of the following represents in terms of ?

Given:

A.
B.
C.
D.
E.

Try to solve it, and check your work!

(spoiler)

Answer: C.

Let’s simplify the expression step by step.

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

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

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

Be careful when reading these questions! Note that is not the same as .

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

This problem is essentially . If you combine the pairs of s and s, you will get three s, i.e.: .

Try this GRE question that involves Rule 5 and 6:

Which mixed number represents this fraction?

A.
B.
C.
D.
E.

Give it a try and check your work!

(spoiler)

Answer: B.

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

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

Bringing it all together: question walkthrough video

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

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