2. Unit 1: Exponents
  3. Lesson 4: Negative Bases and Integer Exponents

Zero Exponent


Explore This 1


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Explore This 1 Summary

Given a positive integer base greater than \(1\), as the exponent becomes larger, then the value of the power also becomes larger. A power with a smaller base becomes larger at a slower rate than one with a larger base, but it still grows in value!

For example, compare a base of \(2\) and a base of \(8\), both being raised to the exponent \(7\).

  • \(2^7=128\)
  • \(8^7=2\space097\space152\)

The base of \(8\) produces a much larger value than a base of \(2\)!

Zero Exponent

It can be shown that regardless of the base, an exponent of \(0\) always results in a value of \(1\). Let's look at why.


Example 1

Evaluate \(\dfrac{2^4}{2^4}\) using the exponent quotient rule and using expanded form.

Solution

Exponent Quotient Rule

\[\begin{align*} \frac{2^4}{2^4}&=2^{4-4}\\ &=2^0 \end{align*}\]

 

Expanded Form

\[\begin{align*} \frac{2^4}{2^4}&=\frac{2\times 2\times 2\times 2}{2\times 2\times 2\times 2}\\ &=\frac{16}{16}\\ &=1 \end{align*}\]

Note that these solutions result in different looking answers. Why?

We know that our math is correct using both methods, so it must mean that these two answers are equivalent, \(2^0=1\).

This example would have the same result with a different base (except \(0\)) raised to the exponent \(4\) and would even be the same with an exponent other than \(4\).

 

Rule

For zero exponents, \(x^0=1\), for \(x\neq0\).

The idea of \(0^0\) is more complex and requires advanced mathematics. It might be something that you will see in a future math course!

Check With Your Calculator

Try the following with your calculator:

  1. Choose any base except \(0\).
  2. Raise that base to the exponent \(0\) (If you are choosing a negative base you will need to use brackets.) 

Your calculator should always give you a value of \(1\).

Example 2

Evaluate each of the following:

  1. \(5^0\)
  2. \(m^0\), \(m\ne0\)
  3. \((12345abcd)^0\), \(a,b,c,d\ne0\)
  4. \(12x^0\), \(x\ne0\)
  5. \((-3)^0\)
  6. \(-3^0\)

Solution

Remember the rule, \(x^0=1\), \(x\ne0\).

  1. For \(5^0\),
    • the exponent is \(0\), and
    • the base is \(5\).
    Therefore, \(5^0=1\).
  2. For \(m^0\),
    • the exponent is \(0\), and
    • the base is \(m\).

    Using the rule, \(m^0=1\).

  3. For \((12345abcd)^0\), remember that an exponent has a base found directly to its left. In this example, a bracket is found directly to the left of the exponent. 
    • The exponent is \(0\).
    • The base is \(12345abcd\).

    Since the base is \(12345abcd\), and it is being raised to the exponent \(0\), then \((12345abcd)^0=1\).

  4. For \(12x^0\),
    • the exponent is \(0\), and
    • the base is \(x\), not \(12x\).

    If the base were \(12x\) it would have been written as \((12x)^0\).

    \[\begin{align*} 12x^0&=12(x^0)\\ &=12(1)\\ &=12 \end{align*}\]
  5. For \((-3)^0\),
    • the exponent is \(0\), and
    • the base is \(-3\) since a bracket is directly to the left of the exponent.

    Using the rule, \((-3)^0=1\).

  6. For \(-3^0\),
    • the exponent is \(0\), and
    • the base is \(3\) , not \(-3\) as in the previous example.

    Using the rule,

    \[\begin{align*} -3^0&=-1\times 3^0\\ &=-1\times 1\\ &=-1 \end{align*}\]

Check Your Understanding 2


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