Exponents

The SBC GMAT Files

Exponents
Staff Writer, Test Prep New York/Test Prep San Francisco

Have you ever heard something like, “The Android Market showed exponential growth”?  What does exponential really mean?

It can be shown if you plot it on a graph – with the ‘multiple product’ on the Y axis and the ‘number of times multiplied’ on the X axis, that in a short amount of time the growth was quite large. Literally, exponents can be defined as multiplying a number by itself.

So in the example of 74, the number being multiplied (the ”˜7’ in this case) is called the base. The number of times it is being multiplied is called the power or exponent (in this case, by ”˜4’). Hence, 74 is read as:

7 raised to the power 4 or

7 raised to the exponent 4 or

7 raised to 4

It means 7 multiplied by itself 4 times: the value of which can be expressed as 7 X 7 x 7 x7 = 2,401.

Conversely, if a number is being divided by the same number as itself, the term ‘root’ is used. The third or cubed root of 8 is 2. This means that when 8 is divided by 2 three times, the answer is 2. It is written as 23. Any number to the 2nd power is referred to as being “squared.” Any number to the 3rd power is referred to as being “cubed.”

We recommend that you memorize squares up to 20, and cubes up to 10.
You can start out, especially if your time is limited, memorizing these roots/exponents that appear commonly on GMAT:

Squares Cubes Higher Powers Other Powers
22 = 4 23 = 8 20 = 1 34 = 81
32 = 9 33 = 27 21 = 2
42 = 16 43 = 64 22 = 4
52 = 25 53 = 125 23 = 8
62 = 36 63 = 216 24 = 16
72 = 49 25 = 32
82= 64 26 = 64
92 = 81 27 = 128
102 = 100 28 = 256
112 = 121 29 = 512
122 = 144 210 = 1024
132 = 169
142 = 196
152 = 225
162 = 256*

*162 = 44 = 28

13 Exponent Rules to Memorize:

  1. Any number raised to the power zero is one, even negative numbers.

50 = 1 and -50 = 1

  1. Any number raised to the power one is the number itself.

51 = 5

  1. Fractions that are raised to a power have the opposite result as whole numbers raised to a power: the higher the exponent to a positive fraction, the lower is the number becomes, for example, (¼)4 ¼ raised to the power 4 is smaller than 1/4 raised to the power

2 ((¼)4  =1/64), and ¼2 =1/16; (1/64 < 1/16)

  1. A number taken to a negative exponent becomes a fraction. You put the number in the denominator with a 1 on the top.

2-3  = 1/8 or 1/2x2x2

  1. A negative number taken to an even exponent becomes positive.

(-24) = 16

  1. A negative number taken to an odd exponent remains negative.

(-23) = -8

  1. A number taken to a fractional exponent becomes a root equal to the value of the fraction.

16½ = or 4

  1. Add the exponent when multiplying two powers of the same base.

25 x 23 = 28

  1. Subtract the exponent when dividing two powers of the same base.

25 /23  = 22

  1. Multiply the exponents when you raise a power to a power.

(22)3 = 26  = 64

  1. The power of a product of factors is written by raising each factor to the specified power.

(2 x 4)3 = (23 x 43 ) or (8)3 = 512

  1. The power of a fraction is written by raising the numerator and the denominator to the specified power.

(2/3)4 = (2)4/(3)4  = 16/81

  1. Exponents of different bases must be multiplied out and then combined.

23 x 42 = 8 x 16 = 128

  1. To note: the rules of manipulating exponents can be useful or necessary in questions without variables. When dealing with large integers, often reduction to prime factors (or at least factors) is necessary for manipulation or cancellation of terms. This involves utilizing the rules of exponents, and a fluid grasp of exponents can help make for fast and efficient calculations.

Effective Strategies for Approaching Exponent Questions on the GMAT

1. Engage with questions wisely.
Often you don’t need to actually solve complicated exponentials. If you were told that a certain virus population multiplies 3 times (3x) in one hour and another multiplies 5 times (5x) in one hour, you pretty much know which one is faster. Since they aren’t fractions, but likely whole numbers or a mixed number > 1, we don’t have to consider the fraction getting smaller when the exponent is larger.

2. Manipulate the given data to get the same base when possible. This makes solving for exponents easier.  Breaking bases down to prime factors often helps.

97 * 36 = (33)7 * 36 = 3 (27) * 36 = 3 33

A more likely GMAT question:

1810 = 45 * 3x

(Hint: Different bases → manipulate the information to get same bases)
(3x3x2)10 = (2×2)5 * (3) x

(3)10 * (3)10 * (2)10 = (2)5 * (2)5 * (3) x

(3)20 * (2)10 = (2)10 * (3) x

x = 20

3. Avoid traps!
If two bases are being added, don’t combine the exponents.
For example, 32 + 36 = is NOT 38. You need to calculate each base then add them.

32 + 36 = 9 + 243 = 252

4. Deal with inequality questions as if the inequality sign was an ”˜=’ sign unless you are multiplying or dividing by a negative number on both sides of the inequality sign.
Example 1.

Example 2: Explaining the exception to the rule. Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

 

 

 

— Dividing by -2 requires the flipping of the inequality sign.

5. On exponent multiplication and even division questions, see if you can solve for a ”˜constant’ in the equation. For multiplication, solve for the ”˜ones’ digit as well as the outer most digits. This won’t give you a definitive answer, BUT it will get rid of many answer choices.  For example:

 234

x  52

1XXX”¦8

The ones digit MUST be 8, the farthest left digit, will likely be 1. The answer? 12,168.  Likely answer choices will only have a couple of answers that are 1XXX”¦8

The following can be figured out similarly in exponent questions:

132 + 123 =

The result of a 32  will yield a 9 in the ones place,

while the result of a 23 will yield an 8 in the ones place.

Added together = 7 in the ones place”¦

therefore you need to find an answer that has a 7 in the ones place.

 ***

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