Integer Exponents & the Quotient Rule

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  • 0:03 Integer Exponents
  • 0:53 Positive Integer Exponents
  • 1:33 Negative Integer Exponents
  • 2:23 Zero Exponent
  • 3:08 Quotient Rule
  • 5:19 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn what integer exponents are. We will look at positive integer exponents, negative integer exponents, the zero exponent, and the quotient rule for exponents. At the end of the lesson, you can test your knowledge with a quiz.

Integer Exponents

Integer exponents are exactly what the name implies. They are integers that are exponents. Thus, let's first recall what integers and exponents are. Integers are numbers with no fractional part. They consist of the counting numbers and their negatives, along with the number zero.

Exponents are numbers that we raise another number to. In an exponential expression, the number being raised to the exponent is called the base, and the number we are raising the base to is called the exponent.

Knowing the definition of an integer and of an exponent allows us to understand the definition of an integer exponent. It is a number that is an exponent and an integer. This image shows some examples of integer exponents:

Integer Exponents 2

Positive Integer Exponents

Positive integer exponents are positive integers that are exponents. When we have a positive integer exponent, the exponent is telling us how many times we want to multiply the base by itself. For instance, consider the expression 2 4. This expression has 2 as its base and 4 as its exponent. Therefore, it's telling us to multiply 2 by itself 4 times.

2 x 2 x 2 x 2 = 16, so 2 4 = 16

Here are some more examples of positive integer exponents:

Integer Exponents 3

Negative Integer Exponents

Negative integer exponents differ from the positive integer exponents in that they consist of negative integers. When we raise a base to a negative integer, the negative flips the numerator and denominator of the base, and then the integer tells us how many times to multiply that number by itself. To demonstrate this, suppose we want to calculate 3 -2. The negative would flip the numerator and denominator of 3 to give 1/3, and then we would multiply 1/3 by itself 2 times.

3 -2 = (1/3) 2 = (1/3)(1/3) = 1/9

We can observe some more examples here:

Integer Exponents 7

Zero Exponent

We've considered positive integers and negative integers as exponents, so what's left? That's right, zero! Zero is neither positive nor negative, so we have a separate rule for when we raise a number to the power of zero. This rule is that anytime we raise a number (no matter what number) to the exponent zero, we get 1. Here are a few examples:

  • 20 0 = 1
  • 53 0 = 1
  • 1,387,720 0 = 1

Get the picture? No matter what our base is, if we raise to the exponent zero, we get 1.

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