Number Properties GRE Strategy Guide by Manhattan Prep

Author:Manhattan Prep
Language: eng
Format: epub
Publisher: MG Prep, Inc.

Chapter 5

of

Number Properties

Exponents

In This Chapter…

Wow, That Increased Exponentially!

All About the Base

All About the Exponent

Combining Exponential Terms

Rewriting Bases

Simplifying Exponential Expressions

Rules of Exponents

Common Exponent Errors

Chapter 5

Exponents

The mathematical expression 43 consists of a base (4) and an exponent (3).

The expression is read as “four to the third power.” The base (4) is multiplied by itself as many times as the power requires (3).

Thus, 43 equals 4 × 4 × 4 = 64.

Two exponents have special names: the exponent 2 is called the square, and the exponent 3 is called the cube. For example, 52 can be read as five to the second power, or as five squared (52 = 5 × 5 = 25); 53 can be read as five to the third power, or as five cubed (53 = 5 × 5 × 5 = 125).

Wow, That Increased Exponentially!

Have you ever heard the expression: “Wow, that increased exponentially!”? This phrase captures the essence of exponents. When a positive number greater than 1 increases exponentially, it does not merely increase; it increases a whole lot in a short amount of time.

An important property of exponents is that the greater the exponent, the faster the rate of increase. Consider the following progression:

51 = 5

52 = 25 Increased by 20

53 = 125 Increased by 100

54 = 625 Increased by 500

The important thing to remember is that for positive bases bigger than 1, the greater the exponent, the faster the rate of increase.

All About the Base

The Sign of the Base

The base of an exponential expression may be either positive or negative. With a negative base, simply multiply the negative number as many times as the exponent requires.

For example:

(−4)2 = (−4) × (−4) = 16 (−4)3 = (−4) × (−4) × (−4) = −64

Consider this problem:

If x2 = 16, is x equal to 4?

Your initial inclination is probably to say yes. However, x may not be 4; it may be −4. Thus, you cannot answer the question without additional information. You must be told that x is positive in order to affirm that x is 4. Beware whenever you see an even exponent on the test. Another important thing to remember is that according to the Order of Operations rules, (PEMDAS), exponents have higher precedence than subtraction, so −42 means −(42) = −16, not (−4)2 = 16.

The Even Exponent Is Dangerous: It Hides the Sign of the Base!

One of the GRE's most common tricks involves the even exponent. In many cases, when an integer is raised to a power, the answer keeps the original sign of the base. For example:

32 = 9 (−3)3 = −27 33 = 27

(positive base, (negative base, (positive base,

positive result) negative result) positive result)

However, any base raised to an even power will always result in a positive answer. This is because even if the underlying base is negative, there will be an even number of negative signs in the product, and an even number of negative signs in a product makes the product positive. For example:

32 = 9 (−3)2 = 9 (−3)4 =

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