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 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, 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 x0, you should know that its value is 1. Note, however, that 00 is undefinded.
One
Any base raised to the power of one is equal to itself. For example, 21 = 2, (–67)1 = –67 and x1 = 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, 213 ⁄ 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.
