A key piece of algebraic notation on GMAT and EA quant problems is the exponent. Exponents appear on many kinds of quantitative problems, so fluency with exponents (and radicals) is an indispensable skill for achieving a competitive quant score. Odds are, you already have some idea of what exponents “mean” in algebraic language, but let’s clarify your definition by exploring how exponents relate to the more fundamental operations of addition and multiplication.
The foundations of all arithmetic are the operations of addition and subtraction. We could even say just addition, since subtraction can be notated as the addition of a negative value. Why have I left out multiplication and division? Well, because multiplication is nothing but an efficient way to notate a special case of addition, and division is nothing but multiplication in reverse. The special case of addition is this: when you want to add up a large number of groups that are all the same size. Let’s say you want to know how many eggs are in stock at your local grocery store. You won’t count the eggs one by one; you’ll count the cartons, since you know that each carton contains 10 or 12 eggs, depending if you’re in America or Europe. You could “show your work” for counting the eggs like this:
12 + 12 + 12 + 12 + 12 + 12 . . .
But this would get out of hand. Multiplication was created for just such a job. Instead of writing out the addition of 217 dozens of eggs, you can write this:
(# of eggs) = 217 * 12
This is many times better than stringing together 217 twelves with plus signs, but the outcome is the same.
The relevance to exponents is this: just as multiplication efficiently notates successive additions of the same value, exponents efficiently notate successive multiplications by the same value.
To stay in the realm of our “eggs at the grocery store” scenario, let’s imagine that a local farm starts out with 5 hens and wants to double its egg-laying workforce every year for the next 7 years. We could notate the target number of hens at the end of the seventh year like this:
5 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 640
But this is a bit impractical. In scenarios where the population of a bacteria doubles hundreds of times, notating with multiplication simply won’t do. We need a better tool, and the tool is exponents. Returning to our hen population example, exponents work like this:
5 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 640
5 * 2⁷ = 640
Rather than stringing together seven 2s with multiplication signs, we can place a 7 as an exponent of the 2 to notate the same thing. When we do this, the 2 is called the base of the exponential expression. The value represented by xⁿ is called the nth power of x or “x to the nth power. In the latter option, “power” is often left tacit, so in our hen scenario, we would verbalize the value of the final population as “five times two to the seventh (power)” or, for a slight simplification, “five times two to the seven” The terms squared and cubed are used for exponents of 2 and 3, respectively. 6² is “six squared”; 5³ is “5 cubed.”
This proper understanding of exponents as shorthand for multiplication makes sense of their properties. Many an algebra student has been tripped up by expressions like this:
x⁷ * x⁴
Seeing the multiplication sign, a novice might incorrectly infer that x⁷ * x⁴ = x²⁸ and be confused by the correction that the exponents should be added, not multiplied, yielding x¹¹. Breaking down the exponential expressions x⁷ and x⁴ to their “original” multiplicative forms should add clarity.
x⁷= x * x * x * x * x * x * x
x⁴ = x * x * x * x
x⁷ * x⁴ = (x * x * x * x * x * x * x) * (x * x * x * x) = x¹¹
Now we can see that the multiplication of the exponential expressions x⁷ and x⁴ is nothing but a chain of multiplications of the variable x: 11 of them, to be exact. And the best way to notate a string of eleven “x’s” in multiplication is with an exponent of 11.
The product of equal bases with different exponents is the base raised to the sum of the exponents. xᵃ * xᵇ = xᵃ⁺ᵇ
With this rule in place, it follows that it can be reversed by “splitting” an exponential expression into two groups.
x¹¹ = x⁷ * x⁴ = x⁶ * x⁵ = x¹⁰ * x
As shown, the way you “split” your expression is flexible. Different algebraic scenarios can benefit from different “splits.”
When the operation is division instead of multiplication, the resulting exponent is calculated via subtraction instead of addition.
x⁷ / x⁴ = (x * x * x * x * x * x * x) / (x * x * x * x)
(x * x * x * x * x * x * x) / (x * x * x * x) = x³
The quotient of equal bases with different exponents is the base raised to the difference of the exponents. xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾
This covers multiplication and division of equal bases with different exponents. Simple rules also exist for multiplication and division of different bases with equal exponents.
x³ * y³ = (xy)³
Again, breaking down the exponential expressions to their “original” multiplicative forms shows why this works:
x³ * y³ = (x * x * x) * (y * y * y)
Everything here is in multiplication, so we can reorder and regroup the factors any way we like.
(x * x * x) * (y * y * y) = xy * xy * xy = (xy)³
Don’t forget the parentheses around your base. Note that we need the parentheses to group the xy as a unit, as opposed to xy³ = x * y*y*y.
Of course, this rule works in reverse as well.
(xy)³ = x³ * y³
Again, different algebraic scenarios call for different algebraic solutions. Both the combination and the “splitting” of algebraic expressions are useful tools in different contexts.
xᵃ * yᵃ = (xy)ᵃ
(xy)ᵃ = xᵃ* yᵃ
As you probably guessed, the same rule applies for division.
x³/ y³ = (x/y)³
And the “proof:”
x³ / y³ = (x * x * x) / (y * y * y) = (x/y) * (x/y) * (x/y) = (x/y)³
And finally, the generalized form of the rule, accompanied by the reversal:
xᵃ / yᵃ = (x/y)ᵃ
(x/y)ᵃ = xᵃ/ yᵃ
One more rule remains to be covered in this introduction. To preview it, let’s return to our idea of “splitting” an exponential expression into pieces:
x¹¹ = x⁷ * x⁴ = x⁶ * x⁵ = x¹⁰ * x
No one said that we have to limit ourselves to two “pieces.” We can keep “splitting” as many times as we want.
x¹¹ = x⁸ * x³ = x⁴ * x⁴ * x³ = x² * x² * x² * x² * x³
Here we see an x³ term multiplied by a string of four “x²” terms. But isn’t there a more efficient way to notate such a string of multiplications? Yes, with exponents! An exponential expression itself can become the base of another exponent.
x³ * x² * x² * x² * x² = x³ * (x²)⁴
Remember that our (x²)⁴ term started out as x⁸. This reveals the rule for simplifying “nested” exponential expressions, or what we call a “power to a power”:
(xᵃ)ᵇ = xᵃ * ᵇ
xᵃ * ᵇ = (xᵃ)ᵇ
This rule makes sense when you know that exponential expressions are “made of” successive multiplications. Four groups of two “x’s” in multiplication is the same thing as 8 “x’s” in multiplication. And you know the drill: the reversal of the rule – where a single exponent is factored to create a “nested” expression – is just as useful as the “original” version.
Now to assemble all of our rules:
xᵃ * xᵇ = x⁽ᵃ⁺ᵇ⁾
x⁽ᵃ⁺ᵇ⁾ = xᵃ + xᵇ
xᵃ / xᵇ = x⁽ᵃ⁻ᵇ⁾
x⁽ᵃ⁻ᵇ⁾ = xᵃ/ xᵇ
xᵃ * yᵃ = (xy)ᵃ
(xy)ᵃ = xᵃ * yᵃ
xᵃ / yᵃ = (x/y)ᵃ
(x/y)ᵃ = xᵃ/ yᵃ
(xᵃ)ᵇ = x⁽ᵃᵇ⁾
x⁽ᵃᵇ⁾ = (xᵃ)ᵇ
And before we try a few official GMAT problems, let’s take a look at some powers of integers you should know:
The main reason for knowing these powers is for something I call “backwards recognition.” If you don’t memorize these and you need to evaluate 54 or 27 in order to solve a problem, you can probably multiply your way through the powers easily enough. But it’s another thing to see 625 or 128 in a problem and immediately know “that’s 54” or “that’s 27.” Such backwards recognition can help you make sense of problems that may look confusing at first.
As a final reminder of the “power to a power” rule, powers of 4 are left out of this list because they are contained within the powers of 2. Every even power of 2 is also a power of 4. For example, 16 = 24 = 42, 64 = 26 = 43, so on and so forth.
Let’s try some official GMAT problems involving exponents. Here’s a simple one to get you started:
216 is
- 2 more than 2¹⁵
- 16 more than 2¹⁵
- ½ of 2³²
- 2 times 2⁸
- 2 times 2¹⁵
If you’ve gotten your head around exponents, this is a 15-second problem. Notice that answer choices C and D both imply the same incorrect rule: that doubling the exponent on the 2 doubles the overall value. Even if you don’t know your exponent rules, you could eliminate these answer choices on logic alone because they are indistinguishable. Answer B doesn’t make much sense, and answer A confuses the rules of exponents with the rules of multiplication. (2 * 16) is 2 more than (2 * 15), but the difference between 216 and 215 is much greater. Answer E gets it right. Increasing the value of the exponent by 1 means to multiply by the base one more time.
Here’s another:
At the start of an experiment, a certain population consisted of 3 animals. At the end of each month after the start of the experiment, the population was double its size at the beginning of that month. Which of the following represents the population size at the end of the 10 months?
- 2³
- 3²
- 2 * 3¹⁰
- 3 * 2¹⁰
- 3 * 10²
If you understood our example with the hens, this should be another easy one. We need to start with 3 and then double our value (multiply by 2) 10 times. Exponents enable us to notate this series of multiplications as 3 * 2¹⁰, answer choice D.
Now for some data sufficiency:
What is the value of 6ˣ 6ʸ ?
- 2⁽ˣ⁺ʸ⁾ = 32
- 3⁽ˣ⁺ʸ⁾ = 243
If you know your exponent rules, you should immediately note that 6x6y can be alternatively written as 6ˣ⁺ʸ. Since this is DS, you don’t necessarily need to know which power of 2 equals 32 or which power of 3 equals 243; it’s enough to recognize that each statement on its own locks in the value of the exponent “x + y” and therefore the value of the 6ˣ⁺ʸ expression that we were asked about. (But ideally you will study your powers enough to know right away that 32 = 2⁵ and 243 = 3⁵.) The correct answer choice is D.
Here’s our final problem for this article:
What is the smallest integer n for which 25ⁿ > 5¹²?
- 6
- 7
- 8
- 9
- 10
This is a case where backwards recognition of powers makes all the difference. 25 should be immediately recognizable as 52. Therefore the given inequality can be rewritten in either of the following ways:
(5²)ⁿ > 5¹²
25ⁿ > 5⁽²×⁶⁾ . . . 25ⁿ > (5²)⁶ . . . 25ⁿ > 25⁶
The second way gets you to the correct answer more quickly, but the first is rather more intuitive. The “power to a power” rule states that the exponents should be multiplied:
(5²)ⁿ > 5¹²
5²ⁿ > 5¹²
Plugging in a 6 makes the exponents equal (2 * 6 = 12), and since the bases are now equal, the expressions would be equal as well. Therefore the smallest integer that works is 7, answer choice B.
I hope you’ve enjoyed this intro to (or review of) exponent properties. As you will definitely come across exponents on the GMAT test, we think that this article is something you need to read carefully! Next time we’ll learn how to “undo” exponents with an inverse operation.