Exponents: Basic Rules

This process of using exponents is called "raising to a power", where the exponent is the "power". The expression 5³ is pronounced as "five, raised to the third power", "five, raised to the power three", or "five to the third".

There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So 5³ is commonly pronounced as "five cubed".

When we deal with numbers, we usually just simplify; we'd rather deal with 27 than with 3³. But with variables, we need the exponents, because we'd rather deal with x⁶ than with x‍x‍x‍x‍x‍x.

What are the rules (or laws) for exponents?

The rules for simplifying with exponents are as follows:

• Product property: ( x^m ) ( xⁿ ) = x^{m + n}
• Power of a power property: ( x^m )ⁿ = x^{m × n}
• Power of a product property: (xy)^m = x^my^m
• Power of a quotient property:
• Quotient property:
• Zero power property: x⁰ = 1
• Negative power property:

Now, what do these rules mean? In what follows, I will illustrate each rule, so you can see how and why the rules work. Once you understand the "why", it's usually pretty easy to remember the "how".

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• Simplify (x³)(x⁴)

Note that x⁷ also equals x⁽³⁺⁴⁾. This demonstrates the first basic exponent rule (that is, the product property):

Note, however, that we can NOT simplify (x⁴)(y³) by adding the exponents, because the bases are different: (x⁴)(y³) = x‍x‍x‍xyyy = (x⁴)(y³). Nothing combines.

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• Simplify (a⁵ b³) (a b⁷)

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In the following example, there are two powers, with one power being, in a sense, "inside" the other.

• Simplify (x²)⁴

Note that (x²)⁴ = x⁸, and that 2 × 4 = 8. This demonstrates the second exponent rule (that is, the power of a power property):

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If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance:

This illustrates the third exponent rule (that is, the power of a product property):

Note: This rule does NOT work if you have a sum or difference within the parentheses. Exponents, unlike mulitiplication, do NOT "distribute" over addition.

For instance, given (3 + 4)², do NOT succumb to the temptation to say, "Hey, this equals 3² + 4² = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)² = (7)² = 49, not 25.

When in doubt, write out the expression according to the definition of the power. For instance, given (x − 2)², don't try to do this in your head. Instead, write it out; "squared" means "multiplying two copies of", so:

The mistake of erroneously trying to "distribute" the exponent is most often made when students are trying to do everything in their heads, instead of showing their work. Do things neatly, and you won't be as likely to make this mistake.

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• Simplify (a² b³ c)⁴

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Just as exponents can distribute over multiplication, they can also distribute over division.

• Simplify

This illustrates the fourth exponent rule (that is, the power of a quotient rule):

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Just as we can multiply two factors with the same base, so we can divide two terms with the same base.

• Simplify

Note that , and that 5 − 2 = 3. This illustrates the fifth power rule (that is, the quotient rule):

(Yes, this rule can lead to negative exponents. You'll learn how to deal with them on the next page.)

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There is one other rule that may or may not be covered in your class at this stage:

This rule is explained on the next page. In practice, though, this rule means that some exercises may be a lot easier than they may at first appear:

• Simplify [(3x⁴y⁷z¹²)⁵ (−5x⁹y³z⁴)²]⁰

By the way, as soon as your class does cover "to the zero power", you should expect an exercise like the one above on the next test. It's a common trick question, designed to make you waste a lot of your limited time — but it only works if you're not paying attention.