Use exponent rules and common bases

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Lesson progressPractice problems 0/3

Difficulty: Beginner
Estimated time: 25 minutes
Domains: Advanced Math
Techniques: Exponent-rulesNegative-exponentsPowers-of-productsCommon-baseExact-equivalence

What you’ll learn

Combine powers that share a base.
Apply an outside exponent to every factor, including the number in front.
Read zero and negative exponents correctly.
Rewrite numbers like $9$ , $27$ , and $81$ with one common base.
Decide when to start with full-curve overlap in Desmos and when to start with the exact exponent rules.

Prerequisites

Distribute, combine, and rewrite expressions0%
Recognize equivalent expressions and distinguish distribution from exponent work.

Why this matters on the SAT

Turn disguised powers into one exact form

SAT exponent questions like to hide a few familiar rules inside messy products, fractions, and powers of numbers. The job is always the same. See which base each exponent belongs to, rewrite the bases so they match, and keep the answer exact. The common mistakes happen in three spots: the number in front, negative exponents, and exponents outside parentheses.

Most of these questions ask which expression is equivalent to the one given. Two expressions are equivalent when they have the same value for every allowed value of their variables. They're two ways of writing the same thing.

Solution to the example

First, list everything inside the parentheses: the number $2$ , then $a^3$ , then $b^{-2}$ . The exponent outside goes to every one of them, so square the $2$ and multiply each exponent by $2$ :

(2a^3b^{-2})^2 =2^2a^{3(2)}b^{-2(2)} =4a^6b^{-4} =\frac{4a^6}{b^4}.

In the last step, $b^{-4}$ moves to the denominator as $b^4$ , because a negative exponent means a reciprocal.

The answer is C. Each wrong choice comes from one of those three spots. Choice A adds $3+2$ where the exponents should multiply. Choice B forgets to square the $2$ in front. Choice D drops the negative sign on $b$ 's exponent instead of moving $b^4$ to the denominator.

SAT example

Which expression is equivalent to

(2a^3b^{-2})^2,

where $a>0$ and $b>0$ ?

A
$\dfrac{4a^5}{b^4}$
B
$\dfrac{2a^6}{b^4}$
C
$\dfrac{4a^6}{b^4}$
D
$4a^6b^4$

Combine only matching bases

In $a^n$ , the $a$ is the base and the $n$ is the exponent. When $n$ is a positive integer, the exponent counts how many copies of the base are multiplied together:

a^4=a\cdot a\cdot a\cdot a.

The main rules come straight from that counting. Multiply two powers of the same base, and you're joining the copies:

a^2a^3 =(a\cdot a)(a\cdot a\cdot a) =a^5.

Two copies plus three copies is five copies, so the exponents add. Now raise a power to another power:

(a^2)^3 =a^2\cdot a^2\cdot a^2 =a^6.

That's three groups of two copies, so the exponents multiply. Here are all the rules in one place. In this table, $m$ and $n$ are integers, and every expression is defined.

Situation	Exact rewrite	What happens to the exponents
Multiply the same base	$a^m a^n=a^{m+n}$	Add them
Divide the same base	$\dfrac{a^m}{a^n}=a^{m-n}$ , $a\ne0$	Subtract the bottom exponent from the top one
Raise a power to a power	$(a^m)^n=a^{mn}$	Multiply them
Raise a product to a power	$(ab)^n=a^n b^n$	Give the exponent to every factor
Use a zero exponent	$a^0=1$ , $a\ne0$	The value is $1$
Use a negative exponent	$a^{-n}=\dfrac1{a^n}$ , $a\ne0$	Take the reciprocal

Negative exponents follow the same rules. For example,

x^5x^{-2}=x^{5+(-2)}=x^3,\qquad x\ne0.

Here's the part that trips people up. A negative exponent doesn't make the value negative. It moves the power into a reciprocal:

x^{-3}=\frac1{x^3},\qquad x\ne0.

Why a reciprocal? Divide $x^2$ by $x^5$ two ways. The quotient rule gives $x^{2-5}=x^{-3}$ . Canceling the two copies of $x$ on top against two on the bottom leaves $\frac1{x^3}$ . Both answers are right, so they have to be equal:

\frac{x^2}{x^5}=x^{2-5}=x^{-3} \quad\text{and}\quad \frac{x^2}{x^5}=\frac1{x^3}, \qquad x\ne0.

Why keep writing $x\ne0$ ? Every form here divides by a power of $x$ . The fraction $\frac{x^2}{x^5}$ does, and $x^{-3}$ means $\frac1{x^3}$ . You can't divide by $0$ , so $x$ can't be $0$ anywhere in this work.

Check your understanding:

For $x\ne0$ , simplify $\dfrac{x^7}{x^7}+(3x)^0$ .

Check your understanding:

For $x>0$ , rewrite $\dfrac{x^{-2}x^7}{x^3}$ using one positive exponent. What is the result?

Common mistake:

If $2^3\cdot4^2$ turns into $2^5$ , the exponents were added before the bases matched. Rewrite $4^2=(2^2)^2=2^4$ first. Then $2^3\cdot2^4=2^7$ . You can check it with plain numbers: the original and $2^7$ both equal $128$ , while $2^5$ is only $32$ .

An exponent spreads across multiplication, not across addition. So $(ab)^2=a^2b^2$ , but $(a+b)^2$ isn't $a^2+b^2$ . Try $a=b=1$ : $(1+1)^2=4$ , while $1^2+1^2=2$ . For powers of sums, use the methods in Distribute, combine, and rewrite expressions.

Example: Track every factor and exponent

Worked example

Which expression is equivalent to

\frac{(3x^{-2}y^4)^3}{9xy^{-1}},

where $x>0$ and $y>0$ ?

A
$\dfrac{3y^{11}}{x^5}$
B
$3x^7y^{13}$
C
$\dfrac{18y^{13}}{x^7}$
D
$\dfrac{3y^{13}}{x^7}$

Step 1

Apply the outside exponent

Start with the top. The exponent $3$ goes to all three factors inside: the $3$ in front, $x^{-2}$ , and $y^4$ .

(3x^{-2}y^4)^3 =3^3x^{-6}y^{12} =27x^{-6}y^{12}.

Step 2

Divide matching factors

Now divide, one kind of factor at a time. The numbers divide as usual, $27\div9=3$ . Each variable subtracts its bottom exponent from its top one. The $x$ on the bottom has no written exponent, so it counts as $x^1$ :

\frac{27x^{-6}y^{12}}{9xy^{-1}} =3x^{-6-1}y^{12-(-1)} =3x^{-7}y^{13}.

Go slowly with $y$ . Its bottom exponent is $-1$ , and subtracting $-1$ is the same as adding $1$ , so $12-(-1)=13$ .

Step 3

Write the final form with positive exponents

The negative exponent on $x$ means a reciprocal, so move $x^7$ to the denominator:

3x^{-7}y^{13} =\frac{3y^{13}}{x^7}.

The answer is D. Choice A adds the bottom exponents instead of subtracting them, and choice C subtracts the numbers, $27-9=18$ , instead of dividing.

Common mistake:

Choice B comes from deleting the minus sign on $x^{-7}$ instead of taking the reciprocal. Keep $x^{-7}$ as it is until the last line, then rewrite it as $\frac1{x^7}$ . To check an answer, multiply it by $x^7$ . What’s left on top should be $3y^{13}$ .

Rewrite numbers with one base

The rules only combine powers of the same base, as the $2^3\cdot4^2$ mistake showed. So when different numbers show up, look for one base they all come from. For example, $9$ , $27$ , and $81$ are all powers of $3$ :

9=3^2,\qquad 27=3^3,\qquad 81=3^4.

When the exponent has a variable in it, the new exponent multiplies the whole thing:

9^{t+1} =(3^2)^{t+1} =3^{2(t+1)} =3^{2t+2}.

Writing $3^{2t+1}$ is a common slip. The $2$ multiplies both terms of $t+1$ , not only the $t$ .

With several powers, work in two passes. First rewrite every base on one line, then do the exponent arithmetic on the next. That way you can see that each power of a power got multiplied and that every exponent in the denominator got subtracted.

Check your understanding:

Rewrite $8^{t+1}\cdot4^{2-t}$ as one power of $2$ .

Pick your method before any long algebra

Some SAT exponent questions are long: one variable, four answer choices, and a pile of powers to rewrite. Converting all of them by hand takes many steps, and one slip anywhere gives a wrong answer. For these, start with full-curve overlap in Desmos. You graph the original and one answer choice, then check whether the two curves sit exactly on top of each other. Desmos draws its curves in x, so change the question's variable to lowercase x on every line:

Enter the whole original expression on line 1.
Enter one answer choice on line 2.
Hide line 2 and show it again by clicking its colored circle. Watch the curve across the whole window.
If the curves separate anywhere, put the next choice on line 2 instead. Stop when a choice stays right on top of the original.

Try it in the calculator. Line 1 is $8^{t+1}\cdot4^{2-t}$ and line 2 is $2^{t+7}$ , both with x in place of $t$ . Hide and show line 2. The curve doesn't move, so this choice matches the original across the whole window.

Look at the whole curve, not one spot. Two different curves can cross at a single point, so one shared point is never enough. Equivalent expressions give the same curve everywhere.

For everything else, start with the exact exponent rules. They're the better choice in four cases:

If one or two steps settle it, like $x^5x^{-2}=x^3$ , working by hand is faster than typing it in.
If more than one variable is left, like the $a$ and $b$ in the SAT example, use the rules.
If you have to write the answer yourself, like the $k$ in $a^k$ , the rules give it to you exactly.
If zero or negative exponents come with a restriction like $x\ne0$ , use the rules so the restriction stays part of your answer.

That last case needs a closer look. Two expressions can match at every input except one. For example, $\frac{x^2}{x}$ and $x$ draw the same line, but $\frac{x^2}{x}$ isn't defined at $x=0$ . On a graph, one missing point is easy to miss. So keep every condition the question gives, like $x>0$ or $x\ne0$ , and check restrictions with algebra, not with the picture.

The algebra confirms this match exactly. It's the same rewrite you did in the last quick check:

8^{t+1}\cdot4^{2-t} =2^{3(t+1)}2^{2(2-t)} =2^{t+7}.

Related: For more practice with the calculator steps, see Equivalent expressions by graph overlap.

Check your understanding:

Which method would you start with for $(3a^2b^{-1})^2$ ? And which for a one-variable question with four long answer choices built from powers of $3$ , $9$ , $27$ , and $81$ ?

Calculator loads as you approach

Line 1 is the original, and line 2 is the choice you’re testing. Hide and show line 2 to see whether the curve changes.

Practice problems

Before you start each problem, look at its length and its variables, and pick your method.

Find the final exponent

Practice problem

For $a>0$ , the expression

\frac{a^9a^{-4}}{a^2}

is equivalent to $a^k$ , where $k$ is a constant. What is the value of $k$ ?

Your answer

Calculator loads as you approach

It’s short, so use the exact rules. To check afterward, change

a

to lowercase x, then graph the original and

x^k

and compare them on the positive side.

Apply a negative outside exponent

Practice problem

Which expression is equivalent to

\left(\frac12x^{-3}y^2\right)^{-2},

where $x>0$ and $y>0$ ?

Calculator loads as you approach

Two variables, so work this one by hand. For an optional arithmetic check, type the original and your answer with

x=2

and

y=3

plugged in, each on its own line. One matching input catches arithmetic slips, but it can’t prove the two forms are equivalent.

Rewrite every factor with one common base

Practice problem

Which expression is equivalent to

\frac{54\cdot9^{2t-1}\cdot27^{3-t}} {2\cdot3^{t+2}\cdot81^{t-2}}

for all real values of $t$ ?

Calculator loads as you approach

One variable, four long choices: a good fit for full-curve overlap.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

For the same base, add exponents to multiply, subtract to divide, and multiply for a power of a power.
An outside exponent goes to every factor, including the number in front.
A zero exponent gives $1$ for a nonzero base, and a negative exponent means a reciprocal, not a negative number.
Rewrite different numbers with one common base before you combine their exponents.
For a long question with one variable and four answer choices to test, start with full-curve overlap.
For short work, more than one variable, an answer you write yourself, or restrictions like $x\ne0$ , use the exact rules and keep every condition the question gives.

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