


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. There are some other terms that you should be familiar with:
 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 (or power) is the 5 in 3^{5}. The exponent tells how many times the base is to be multiplied by itself.
 Square. Saying that a number is “squared” means that it 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 Exponents
It may be worth your while to memorize a few common exponents
before the test. Knowing these regularly used exponents can save
you the time it would take to calculate them during the test. Here
is a list of squares from 1 through 10:
Memorizing the first few cubes can be helpful as well:
Finally, the first few powers of two are useful for many
applications:
Adding and Subtracting Numbers with Exponents
In order to add or subtract numbers with exponents, you
have to first find the value of each power, and 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, finally, 27 + 16 =
43.
If you’re dealing with algebraic expressions that have
the same bases and exponents, such as 3x^{4} and
5x^{4}, then they can simply
be added and subtracted. For example, 3x^{4} +
5x^{4} = 8x^{4}.
Multiplying and Dividing Numbers with Exponents
To multiply exponential numbers or terms that have the
same base, add the exponents together:
To divide two samebase exponential numbers or terms,
just subtract the 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:
If you need to multiply or divide two exponential numbers
that do not 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 the result accordingly.
Raising an Exponent to an Exponent
Occasionally you might encounter an exponent raised to
another exponent, as seen in the following formats (3^{2})^{4} and
(x^{4})^{3}.
In such cases, multiply the powers:
Exponents and Fractions
To raise a fraction to an exponent, raise both the numerator
and denominator to that exponent:
Exponents and Negative Numbers
As we said in the section on negative numbers, when you
multiply a negative number by another 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 evennumber 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, all you have to do is look at 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 potentially 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 need to 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
undefinded.
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:
Or, 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 , is called the radicand.
For a more familiar example, look at 9^{1⁄2},
which is the same as :
Fractional exponents will play a large role on SAT II
Math IC, so we are just giving you a quick introduction to the topic
now. Don’t worry if some of this doesn’t quite make sense now; we’ll
go over roots thoroughly in the next section.
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:
With that, you’ve got the four rules of special exponents.
Here are some examples to firm up your knowledge:
