Introduction

This video illustrates the lesson material below. Watching the video is optional.

• Rules of Exponents: Exponents of 0 and 1 (04:16 mins) | Transcript

Exponents of 1

Example 1
\begin{align*} \frac{a^4}{a^3} = a^{4-3} = a^{1}\end{align*}

Because \(a^{1}=a\), the answer is a.

Another way to simplify the expression \(\large\frac{a^4}{a^3}\) is by canceling like bases from the numerator and denominator like this:

\begin{align*} \large\frac{a^{4}}{a^{3}} = \frac{{\color{red}\cancel a \color{red} \cdot \cancel a\cdot \color{red}\cancel a}\cdot a}{\color{red}\cancel a\cdot \cancel a\cdot \cancel a} = a \end{align*}

So again, the answer is \(a^{1}\), or a.

Anything raised to the one power is just itself.

Exponents of 0

Any base, like \(a\), raised to the power of 0, \(a^0\), is equal to 1. So \(a^{0}=1\).

Example 2
\begin{align*} \frac{a^{2}}{a^{2}}=a^{2-2}=a^{0}\end{align*}

To understand what this means, simplify the same expression by canceling the common bases.
\begin{align*} \frac{a^{2}}{a^{2}} = \frac{\color{red}\cancel a\cdot \cancel a}{\color{red}\cancel a\cdot \cancel a} = 1\cdot 1 = 1 \end{align*}

\(\frac{a^{2}}{a^{2}}=a^{0} =1\). This is true for any value for \(a\), except for zero, \(0^{0}\neq1\). Base 0, to the power of 0 is undefined.

Example 3
Calculate \(3^{0}\).

According to the rule of exponents of 0, \(3^{0}=1\).

Think of a way that you can have a zero power exponent. It could be a scenario such as \(3^{4-4}\). This is the same as \(\large\frac{3^{4}}{3^{4}}\). If you expand this to see how to simplify it, it looks like this:
\begin{align*}
\frac{3^{4}}{3^{4}} = \frac{\color{red}\cancel 3\cdot \cancel 3\cdot \cancel 3\cdot \cancel 3}{\color{red}\cancel 3\cdot \cancel 3\cdot \cancel3\cdot \cancel 3} = 1 \cdot 1 \cdot 1 \cdot 1 = 1 \end{align*}

Once again, each of these can divide each other out. It just becomes one times one times one times one, which equals one.

And that’s true for any number, \(8^{0}=1\), \(25673^{0} = 1\), or anything raised to the zero power is one.

Things to Remember

• Any number to the 1 power is just the number itself. 
• Any number to the 0 power is 1. (Remember that the only exception to this rule is 0 raised to the 0 power, \(0^0 \neq 1\).

Practice Problems

1. \(3^{1}=?\) (
   Solution

   x

   Solution: 3
   Details:
   Any number raised to the power of 1 is still itself, so \(3^{1}\) is just 3.
   
   This is because the exponent means the number of 3’s being multiplied together.
   )
2. \({\text{y}}^{1}=?\) (
   Solution

   x

   Solution: y
   )
3. \(5^{0}=?\) (
   Solution

   x

   Solution: 1
   Details:
   Any number raised to the power of 0 is 1.
   
   The following explains why:
   
   You can prove this using what you know about division and the quotient rule. First, you know that anything divided by itself equals 1. For example, \(\dfrac{7}{7}=1\) or \(\dfrac{\text{m}}{\text{m}}= 1\). Therefore, something like \(\dfrac{5^{1}}{5^{1}}=\dfrac{5}{5}=1\) since this is 5 divided by itself. Using the quotient rule, you also know that \(\dfrac{5^{1}}{5^{1}}=5^{(1-1)}=5^{0}\). The rest is a matter of logic. If \(\dfrac{5^{1}}{5^{1}}=1\) and \(\dfrac{5^{1}}{5^{1}} =5^{0}\), then \(5^{0} = 1\) as well.
   )
4. \({\text{b}}^{0}=?\) (
   Solution

   x

   Solution: 1
   )
5. \(6^{(8-8)}=?\) (
   Video Solution

   x

   Solution: 1
   Details:
   

   
   (Exponents 0 and 1 #5 (01:25 mins) | Transcript)
   | Transcript)
6. \(\dfrac{{\text{x}}^{9}}{{\text{x}}^{9}}=?\) (
   Video Solution

   x

   Solution: 1
   Details:
   

   
   (Exponents 0 and 1 #6 (01:05 mins) | Transcript)
   | Transcript)

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