Exponents: Basic Rules

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".

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" 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".

Exponents have a few rules that we can use for simplifying expressions.

Simplify (x3)(x4)   Copyright © Elizabeth Stapel 2000-2011 All Rights

To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form:

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

Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x m ) ( x n ) = x( m + n )

However, we can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines.

Simplify (x2)4

Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the fourth" means that I'm multiplying four copies of x2:

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

Note that x8 also equals x( 2×4 ). This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

( xm ) 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:

Warning: 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 be covered at this stage:

Anything to the power zero is just "1".

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 don't, because the zero power on the outside means that the value of the entire thing is just 1.