


Exponents
An exponent defines the number of times a number is to
be multiplied by itself. For example, in a^{b},
where a is the base and b the
exponent, a is multiplied by itself b times.
In a numerical example, 2^{5} = 2 2 2 2 2.
An exponent can also be referred to as a power: a number with an
exponent of 2 is raised to the second power. The following are other terms
related to exponents with which you should be familiar:
 Base. The base refers to the 3 in 3^{5}. It is the number that is being multiplied by itself however many times specified by the exponent.
 Exponent. The exponent is the 5 in 3^{5}. It indicates the number of times the base is to be multiplied with itself.
 Square. Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 6^{2}, 6 has been squared.
 Cube. Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 4^{3}, 4 has been cubed.
Common Powers
It may be worthwhile to memorize a few common powers before
taking the Math IIC, in order to save the time you’d take to calculate
them during the test. Here is a list of squares from 1 through 10.
Memorizing the first few cubes might also be helpful:
The first few powers of 2 are also useful to know for
many applications:
Adding and Subtracting Numbers with Exponents
In order to add or subtract numbers with exponents, you
must first find the value of each power, then add the two numbers.
For example, to add 3^{3} + 4^{2},
you must expand the exponents to get (3 3 3) + (4 4), and then, 27 + 16 = 43.
However, algebraic expressions that have the same bases
and exponents, such as 3x^{4} and 5x^{4},
can be added and subtracted. For example, 3x^{4} +
5x^{4} = 8x^{4}.
Multiplying and Dividing Numbers with Exponents
To multiply exponential numbers raised to the same exponent,
raise their product to that exponent:
To divide exponential numbers raised to the same exponent,
raise their quotient to that exponent:
To multiply exponential numbers or tems that have the
same base, add the exponents together:
To divide two samebase exponential numbers or terms,
just subtract the exponents:
If you need to multiply or divide two exponential numbers
that don’t have the same base or exponent, you’ll just have to do
your work the oldfashioned way: multiply the exponential numbers
out, and multiply or divide them accordingly.
Raising an Exponent to an Exponent
Occasionally you might encounter a power raised to another
power, as in (3^{2})^{4} and
(x^{4})^{3}.
In such cases, multiply the exponents:
Exponents and Fractions
To raise a fraction to an exponent, raise both the numerator
and the denominator to that exponent:
Exponents and Negative Numbers
As we said in the negative numbers section, when you multiply
a negative number by a negative number, you get a positive number,
and when you multiply a negative number by a positive number, you
get a negative number. These rules affect how negative numbers function
in reference to exponents.
 When you raise a negative number to an even number exponent, you get a positive number. For example, (–2)^{4} = 16. To see why this is so, let’s break down the example. (–2)^{4} means –2 –2 –2 –2. When you multiply the first two –2s together, you get 4 because you are multiplying two negative numbers. Then, when you multiply the 4 by the next –2, you get –8, since you are multiplying a positive number by a negative number. Finally, you multiply the –8 by the last –2 and get 16, since you’re once again multiplying two negative numbers.
 When you raise a negative number to an odd power, you get a negative number. To see why, refer to the example above and stop the process at –8, which equals (–2)^{3}.
These rules can help a great deal as you go about eliminating
answer choices and checking potential correct answers. For example,
if you have a negative number raised to an odd power and you get
a positive answer, you know your answer is wrong. Likewise, on that same
question, you could eliminate any answer choices that are positive.
Special Exponents
There are a few special properties of certain exponents
that you also should know.
Zero
Any base raised to the power of zero is equal to 1. If
you see any exponent of the form x^{0}, you
should know that its value is 1. Note, however, that 0^{0} is
undefined.
One
Any base raised to the power of one is equal to itself.
For example, 2^{1} = 2, (–67)^{1} =
–67, and x^{1} = x.
This can be helpful when you’re attempting an operation on exponential
terms with the same base. For example:
Fractional Exponents
Exponents can be fractions, too. When a number or term
is raised to a fractional power, it is called taking the root of
that number or term. This expression can be converted into a more convenient
form:
For example, 2^{13 ⁄ 5} is equal
to the fifth root of 2 to the thirteenth power:
The symbol is also
known as the radical, and anything under the radical (in this case
2^{13}) is called the radicand. For a more
familiar example, look at 9^{1⁄2}, which
is the same as :
Negative Exponents
Seeing a negative number as a power may be a little strange
the first time around. But the principle at work is simple. Any
number or term raised to a negative power is equal to the reciprocal
of that base raised to the opposite power. For example:
Or a slightly more complicated example:
You’ve got the four rules of special exponents. Here are
some examples to firm up your knowledge:
