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Exponents: Basic Rules (page 1 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents

Exponents are shorthand for multiplication: (5)(5) = 5², (5)(5)(5) = 5³. The "exponent" stands for however many times the thing is being multiplied. The thing that's being multiplied is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". "5³" is "five, raised to the third power". 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".

There are a few rules that simplify our dealings with exponents. Given the same base, there are ways that we can simplify various expressions. For instance:

• Simplify (x³)(x⁴)   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Think in terms of what the exponents mean:

(x³)(x⁴) = (xxx)(xxxx)
          = xxxxxxx
          = x⁷

...which also equals x⁽³⁺⁴⁾. This demonstrates a basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x ^m ) ( x ⁿ ) = x^{( m + n )}

Note that we cannot simplify (x⁴)(y³), because the bases are different: (x⁴)(y³) = xxxxyyy = (x⁴)(y³). Nothing combines.

• Simplify (x²)⁴

Again, think in terms of what the exponents mean:

(x²)⁴ = (x²)(x²)(x²)(x²)
       = (xx)(xx)(xx)(xx)
       = xxxxxxxx
       = x⁸

...which also equals x^{( 2×4 )}. This demonstrates another rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

( x^m ) ⁿ = x ^{m n}

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy²)³ = (xy²)(xy²)(xy²) = (xxx)(y²y²y²) = (xxx)(yyyyyy) = x³y⁶ = (x)³(y²)³. Another example would be:

Note that 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 "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. Given (x – 2)², don't try to do this in your head. Instead, write it out: "squared" means "times itself", so (x – 2)² = (x – 2)(x – 2) = xx – 2x – 2x + 4 = x² – 4x + 4.

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

There is one other rule that may or may not arise at this stage:

Anything to the power zero is just "1".

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

Who cares about that stuff inside the square brackets? I know I don't, because the zero power on the outside means that the value of the entire thing is just 1.

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