Exponents
Exponents
An exponent is a shorthand way of saying, “Multiply this number by itself this number of times.” In ab, a is multiplied by itself b times. Here’s a numerical example: 25 = 2 × 2 × 2 × 2 × 2. An exponent can also be referred to as a power: 25 is “two to the fifth power.” Before jumping into the exponent nitty-gritty, learn these five terms:
  • Base. The base refers to the 3 in 35. In other words, the base is the number multiplied by itself however many times specified by the exponent.
  • Exponent. The exponent is the 5 in 35. The exponent tells how many times the base is to be multiplied by itself.
  • Squared. Saying that a number is squared is a common code word to indicate that it has an exponent of 2. In the expression 62, 6 has been squared.
  • Cubed. Saying that a number is cubed means it has an exponent of 3. In the expression 43, 4 has been cubed.
  • Power. The term power is another way to talk about a number being raised to an exponent. A number raised to the third power has an exponent of 3. So 6 raised to the fourth power is 64.
One last word of exponent advice before we get started. We’ll explain how to deal with exponents without using your calculator, but most good graphing calculators have a variety of exponent functions. Knowing how to use them could save you time, especially when exponent values get larger and involve fractions or negative numbers.
Common Exponents
It can be very helpful and a real time-saver on the SAT if you can easily translate back and forth between a number and its exponential form. For instance, if you can easily see that 36 = 62, it can really come in handy if you’re dealing with binomials, quadratic equations, or any number of other areas in algebra.
Here are some lists of common exponents. We’ll start with the squares of the first ten integers:
Here are the first five cubes:
Finally, the first few powers of two are useful to know for various applications:
Adding and Subtracting Powers
Actually, you can’t add or subtract numbers with exponents. Instead, work out each exponent to find its value, then add the two numbers. To add 33 + 42, work out the exponents to get (3 × 3 × 3) + (4 × 4), and then, finally, 27 + 16 = 43.
But if you’re dealing with algebraic expressions that have the same bases and exponents, such as 3x4 and 5x4, then you can add or subtract them. For example, 3x4 + 5x4 = 8x4. Algebraic expressions that have different bases or exponents cannot be added or subtracted.
Multiply and Dividing Powers
Multiplying or dividing exponential numbers or terms that have the same base is so quick and easy it’s like a little math oasis. When multiplying, just add the exponents together. This is known as the Product Rule:
To divide two same-base exponential numbers or terms, subtract the exponents. This is known as the Quotient Rule:
Quick and easy. Right?
You want the bad news or the bad news? The same isn’t true if you need to multiply or divide two exponential numbers that don’t have the same base, such as, say, 33 × 42. When two exponents have different bases, you just have to do your work the old-fashioned way: Multiply the numbers out and multiply or divide the result accordingly: 33 × 42 = 27 + 16 = 43.
Raising a Power to a Power
To raise one exponent to another exponent, multiply the exponents. This is known as the Power Rule:
Again, easy. Just remember that you multiply exponents when raising one exponent to another, and you add exponents when multiplying two identical bases with exponents. The SAT expects lots of students to mix these operations up.
Fractions Raised to a Power
To raise a fraction to an exponent, raise both the numerator and denominator to that exponent:
Negative Numbers Raised to a Power
Here’s a fact you should already know: 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. Since exponents result in multiplication, a negative number raised to an exponent follows these rules:
  • A negative number raised to an even exponent will be positive. For example (–2)4 = 16. Why? Because (–2)4 means –2 × –2 × –2 × –2. When you multiply the first two –2s together, you get positive 4 because you’re multiplying two negative numbers. When you multiply the +4 by the next –2, you get –8, since you’re 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. The negatives cancel themselves out and vanish.
  • A negative number raised to an odd exponent will be negative. To see why, just look at the example above, but stop the process at –23, which equals –8.
Special Exponents
You need to know a few special types of exponents for the SAT.
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. Strange, but true.
One
Any base raised to the power of one is equal to itself: 21 = 2, –671 = –67 and x1 = x. This fact is important to know when you have to multiply or divide exponential terms with the same base:
Negative Exponents
Any number or term raised to a negative power is equal to the reciprocal of that base raised to the opposite power. Uh. Got that? Didn’t think so. An example will make it clearer:
Here’s a more complicated example:
Here’s a translation: If you see a base raised to a negative exponent, put the base as the denominator under a numerator of 1 and then drop the negative from the exponent. From there, just simplify.
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:
The symbol is also known as the radical, and anything under the radical is called the radicand. We’ve got a whole section devoted to roots and radicals coming right up.
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