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

$$3^4\ast 3^5=3^9$$

In this problem, both bases of $3^4$ and $3^5$ are $3$. When you expand both exponents you get the following:

$$3^4\times 3^5=(3\times 3\times 3\times 3)\times (3\times 3\times 3\times 3\times 3)=3^9$$

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

$$3^6/3^3=3^3$$

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

$$3^6/3^3=(3\times 3\times 3\times 3\times 3\times 3)/(3\times 3\times 3)=3\times 3\times 3=3^3$$

Try this GRE problem that involves Rule 1 and 2:

$0<x<1$

Quantity A: $2^4(2^3)x^3/2^5$
Quantity B: $4^4{x}^5/(4x{)}^3$

Do you know the answer?

Let’s simplify Quantity A first:

$$\begin{array}{c}{2}^4(2^3)x^3/2^5\\ 2^7{x}^3/2^5\\ 2^2{x}^3\\ 4x^3\end{array}$$

And let’s simplify Quantity B:

$$\begin{array}{c}{4}^4{x}^5/(4x{)}^3\\ 4^4{x}^5/((4^3)(x^3))\\ 4^4{x}^2/4^3\\ 4^1{x}^2\\ 4x^2\end{array}$$

Now we can compare them side by side:

$$\begin{array}{rl}A:& 4x^3\\ B:& 4x^2\end{array}$$

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

$$3^{-4}=1/(3^4)$$

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 $3^4/3^8$ would be equal to $3^{-4}$ because $4-8=-4$. Because four of the $3$s on the bottom are canceled out by the four of the $3$s on the top, we are left with $1/(3^4)$.

Rule 4: Any number to a 0 exponent is 1

$$3^0=1$$

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 $1$.

$$3^3/3^3=3^{3-3}=3^0=1$$

Try this GRE question that involves Rule 3 and 4:

Which of the following represents $y$ in terms of $z$?

Given: $z^{-4}{z}^4{x}^{-y}=x^y{x}^z$

A. $z$
B. $z/2$
C. $-z/2$
D. $2/z$
E. $-2/z$

Try to solve it, and check your work!

Let’s simplify the expression step by step.

$$\begin{array}{rl}{z}^{-4}{z}^4{x}^{-y}& =x^y{x}^z\\ z^{-4+4}{x}^{-y}& =x^y{x}^z\\ z^0{x}^{-y}& =x^y{x}^z\\ (1)x^{-y}& =x^y{x}^z\\ x^{-y}& =x^y{x}^z\\ 1/x^y& =x^y{x}^z\\ 1& =x^{2y}{x}^z\\ 1/x^z& =x^{2y}\\ x^{-z}& =x^{2y}\\ -z& =2y\\ -z/2& =y\end{array}$$

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

$$\begin{array}{rl}{z}^{-4}{z}^4{x}^{-y}& =x^y{x}^z\\ z^{-4+4}{x}^{-y}& =x^y{x}^z\\ z^0{x}^{-y}& =x^y{x}^z\\ x^{-y}& =x^y{x}^z\\ x^{-y}& =x^{y+z}\\ -y& =y+z\\ 0& =2y+z\\ -z& =2y\\ -z/2& =y\end{array}$$

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

$$(3^3{)}^2=3^6$$

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

Be careful when reading these questions! Note that $(3^3{)}^2$ is not the same as $3^{(3^2)}$.

$$\begin{gathered} (3^3{)}^2=3^6=729 \\ {3}^{(3^2)}=3^9=19683 \end{gathered}$$

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

$$3^3\times 4^3=12^3$$

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

Try this GRE question that involves Rule 5 and 6:

Which mixed number represents this fraction?

$$\frac{3^{2}11^2}{((2^2{)}^2{)}^2}$$

A. $3\frac{25}{256}$
B. $4\frac{65}{256}$
C. $4\frac{155}{256}$
D. $4\frac{235}{256}$
E. $5\frac{65}{256}$

Give it a try and check your work!

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

$$\begin{array}{c}{3}^{2}11^2/((2^2{)}^2{)}^2\\ (3\ast 11{)}^2/((2^2{)}^2{)}^2\\ 33^2/((2^2{)}^2{)}^2\\ 1089/((2^2{)}^2{)}^2\\ 1089/(2^2{)}^{2\ast 2}\\ 1089/(2^2{)}^4\\ 1089/2^{2\ast 4}\\ 1089/2^8\\ 1089/256\\ \approx 4.25390625\end{array}$$

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:

$$\sqrt{25}=\sqrt[2]{25^1}=25^{1/2}$$

Bringing it all together: question walkthrough video

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