Exponential Powers - GMAT Math Study Guide

Definitions

An exponential expression is simply multiplication repeated (e.g., 23=2*2*2).

  • Base - the number that is multiplied by itself a certain quantity of times.
    For example, in the expression 23, the number 2 is the base.
  • Exponent - the number of times a quantity is multiplied by itself.
    For example, in the expression 23, the number 3 is the exponent.
  • Power - a synonym for exponent.
    For example, if one were to say, "raise 3 to the 4th power," the base would be 3 and the exponent (or power to which 3 is raised) would be 4.

Table Clarifying Definitions

The following chart breaks down the parts in an exponential expression, clarifying exactly which number is the exponential power.

Expression Long-Hand Expression Base Exponent Power Value
23 2*2*2 2 3 3 8
46 4*4*4*4*4*4 4 6 6 4096
32 3*3 3 2 2 9
64 6*6*6*6 6 4 4 1296

In reading math problems, expressions with exponential powers such as 32 are often pronounced "three to the second power." Alternatively, exponential expressions such as 32 are often read as "the second power of three."

Rules of Exponents

Expotential Powers Formula
Expotential Powers Formula
Expotential Powers Formula
Expotential Powers Formula
Expotential Powers Formula
Expotential Powers Example
1n = 1
x0 = 1
0n = 0

Note: During the past decade, mathematicians argued extensively about the value of 00. Some answer that 00 = 1 while others answer that 00 is undefined. In the unlikely event that this question appears in some format or is a required intermediary calculation, the correct answer is more likely that 00 = 1

Examples of the Rules of Exponents

Expotential Powers Example
2324 = 23+4 = 27
2434 = (2*3)4 = 64
Expotential Powers Example
Expotential Powers Example
(32)4 = 3(2*4) = 38

Exponential Powers Grow Expressions Rapidly

Exponential powers increase the value of an expression at an incredibly large rate. In order to see this, consider the following example:

21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512

In each instance, the value is doubling--which makes sense since it is being multiplied by 2 another time for each increase in the value of the exponent.

Types of Exponents

Positive Exponents

The most basic and common type of exponent is a positive exponent. The expression xy has a positive exponent if y > 0. The following are all examples of positive exponents.

1020
21
32

Negative Exponents

Although it is most common to see an exponential expression with a base raised to a positive power, a base can just as easily be raised to a negative power. The expression xy has a positive exponent if y < 0. In working with negative exponential powers, it is extremely important to remember the following formula.

Expotential Powers Formula

As a result of this formula, an exponential equation can often be simplified. Consider the following examples:
Note: The first example could be solved using the formula: xnxm = xn+m

Expotential Powers Example
Expotential Powers Example

Fractional Exponents

While many exponential expressions are raised to an integer power, nothing prevents a base from being raised to a fractional power. Although all fractional exponents follow the aforementioned rules and therefore are alike, it is often helpful to break down fractional exponents in a separate lesson since the use of radicals and roots is involved.

Exponent of Zero

Any number raised to the 0 power is one.

x0 = 1

Exponent of 1

Any number raised to the first power is simply that number.

x1 = x

Recursive Exponents

A recursive exponential expression is one in which multiple exponents are nested within each other. For example:

Expotential Powers Example

As per the order of operations, you evaluate an expression such as this by first computing the value inside parenthesis (there are none here) and then performing exponential expressions by working from left to right. Consequently, the expression above is evaluated in the following manner.

1. 22 = 4
2. 43 = 64

In the above example, it would be wrong to first compute 23 = 8 and then compute 28 = 256