Overview:
Exponents, or powers, are shorthand for repeated multiplication of the same number, variable, or expression. They are especially useful to describe variables and expressions that are multiplied over and over again.
How Do Positive Exponents Work?
Positive exponents are a signal to multiply the base that number of times. For example, if the number is 103, 10 should be multiplied as 10 ∙ 10 ∙ 10, or 1000. If the variable is x9, the variable x should be multiplied by itself 9 times, as x∙x∙x∙x∙x∙x∙x∙x∙x. If the expression is (3x)3, then 3x is multiplied by itself 3 times, as 3x∙3x∙3x, or 9x3.
What If the Exponent Is 1 or 0?
If the exponent is 1, that means that the base is multiplied 1 time. In other words, the base is just itself. If 101 is 10, then x1 is x, and (8x)1 is 8x. In algebraic terms, any number n1 equals n. By definition, any number with an exponent of 0 is equal to 1. Therefore, 100 equals 1, (4y)0 equals 1, and z0 equals 1, as long as none of the variables are equal to 0.
What Are Negative Exponents?
Negative exponents are shorthand for the inverse of the number with a positive exponent. For example, 10-2 is 1/102, or 1/100. Similarly, z-4 is 1/z4. For the algebraic definition to work, the number must be a positive real number and it cannot be equal to zero. That is because dividing by zero is undefined.
What About Simplifying Expressions?
When an expression has a negative exponent, it can be simplified by either using the definition of a negative exponent or solving the exponent. For example, y-9 equals 1/y9, just as (-5)-3 is the same thing as -1/53, which can be solved as -1/125. In some equations, it is an advantage to simplify the expression.
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