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Simplifying Expressions With An Exponent of Zero

Simplifying Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like \(\cfrac{{a}^{m}}{{a}^{m}}\). From your earlier work with fractions, you know that:

\(\cfrac{2}{2}=1\phantom{\rule{2em}{0ex}}\cfrac{17}{17}=1\phantom{\rule{2em}{0ex}}\cfrac{-43}{-43}=1\)

In words, a number divided by itself is 1. So, \(\cfrac{x}{x}=1\), for any \(x\phantom{\rule{0.2em}{0ex}}\left(x\ne 0\right)\), since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify \(\cfrac{{a}^{m}}{{a}^{n}}\) when \(m>n\) and when \(n<m\) by subtracting exponents. What if \(m=n\)?

Consider \(\cfrac{8}{8}\), which we know is 1.

\(\begin{array}{cccccc}& & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{8}{8}& =\hfill & 1\hfill \\ \text{Write 8 as}\phantom{\rule{0.2em}{0ex}}{2}^{3}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\cfrac{{2}^{3}}{{2}^{3}}& =\hfill & 1\hfill \\ \text{Subtract exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{3-3}& =\hfill & 1\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{0}& =\hfill & 1\hfill \end{array}\)

Now we will simplify \(\cfrac{{a}^{m}}{{a}^{m}}\) in two ways to lead us to the definition of the zero exponent. In general, for \(a\ne 0\):

This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.

We see \(\cfrac{{a}^{m}}{{a}^{m}}\) simplifies to \({a}^{0}\) and to 1. So \({a}^{0}=1\).

Zero Exponent

If \(a\) is a non-zero number, then \({a}^{0}=1\).

Any nonzero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example

Simplify:

  1. \({9}^{0}\)
  2. \({n}^{0}.\)

Solution

The definition says any non-zero number raised to the zero power is 1.

  1.  

    \(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{9}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)

  2.  

    \(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{n}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let’s look at \({\left(2x\right)}^{0}\). We can use the product to a power rule to rewrite this expression.

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(2x\right)}^{0}\hfill \\ \text{Use the product to a power rule.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{2}^{0}{x}^{0}\hfill \\ \text{Use the zero exponent property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1·1\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)

This tells us that any nonzero expression raised to the zero power is one.

Example

Simplify:

  1. \({\left(5b\right)}^{0}\)
  2. \({\left(-4{a}^{2}b\right)}^{0}.\)

Solution

  1.  

    \(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(5b\right)}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)

  2.  

    \(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(-4{a}^{2}b\right)}^{0}\hfill \\ \text{Use the definition of the zero exponent.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}\)

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