Numbers & Operations
Exponents, or The Powers That Be
The word exponent means “tiny raised number.” That’s not 100% mathematically accurate, but it’s true from a totally visual standpoint.
An exponent above a number tells you how many times that number has to be multiplied by itself. For the number on the left above, the base is 5 and the exponent is 2. This means we must multiply the base of 5 by itself twice:
When the exponent is 2, you can also use the term squared to describe the expression. In the above example, 5 squared is 25. The number 25 is a perfect square because it is the product of an integer squared. The first ten perfect squares are:
When the exponent is 3, you can say the base is being cubed:
So 7 cubed is 343. One other way to talk about exponents is to use the term power when describing the exponent. Five squared can also be called “five to the second power” or “five to the power of two,” and 7 cubed can be called “seven to the third power” or “seven to the power of three.”
Now that you understand the basic terms, let’s start messing with exponents.
Adding and Subtracting Powers: A Big No-No
Numbers with exponents cannot be simply added or subtracted. If you see , it would be easy to say the answer is . It would also be quite wrong:
whereas
To add or subtract numbers with exponents, compute the values of the numbers first, then add:
If the base and the exponent are the same, you can add or subtract expressions. For instance, in the expression , both terms have the same base (the variable y) and the same exponent (4). Therefore, you can express it as:
When algebraic expressions have different bases or different exponents, you cannot add or subtract.
Multiplying and Dividing Powers: Yeah, That’s Allowed
Think about how the term is just shorthand for (s)(s)(s). The term is shorthand for (s)(s)(s)(s). If you multiply these two terms together, you’re really just adding the exponents together:
If you have the same base, you can add exponents together when multiplying. When dividing, you can subtract them, but again, the bases have to be exactly the same:
The Freaky Exponent Stuff
Because the new SAT wants to emphasize some of the more obscure corners of the math world, you can expect the test to set some traps using unusual exponents.
One of these traps comes in the form of raising a power to a power, . This is “three squared then cubed,” for those who like the lingo. To simplify it, all you have to do is multiply the exponents:
If you want to understand why this works, write out all the 3s involved, and you’ll come up with the same answer.
Negative numbers, in both the base and exponents, are another way the SAT tries to trap you. When you raise a negative base number to a power (or multiply it by itself), you may have to change the sign of the number:
A negative number remains negative when raised to an odd exponent:
If you have a negative exponent, first make it positive, then make that new number the denominator of a fraction with 1 as the numerator.
What if the exponent is a fraction? (We told you the test is filled with fractions.) If that’s the case, the numerator becomes the power to which the number is raised, and the denominator is the root of the number.
The symbol is called a radical, and anything under the radical is called a radicand. Here’s an example of a fractional exponent:
Want to know more about roots and radicals? Well, we won’t keep you waiting.
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