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Simplifying Expressions with Zero Exponents

Simplifying Expressions with Zero Exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $$\frac{{a}^{m}}{{a}^{m}}.$$ From earlier work with fractions, we 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 $$\frac{x}{x}=1,$$ for any $$x$$ ($$x\ne 0$$), since any number divided by itself is $$1.$$

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

Now we will simplify $$\frac{{a}^{m}}{{a}^{m}}$$ in two ways to lead us to the definition of the zero exponent.

Consider first $$\frac{8}{8},$$ which we know is $$1.$$

 $$\frac{8}{8}=1$$ Write 8 as $${2}^{3}$$. $$\frac{{2}^{3}}{{2}^{3}}=1$$ Subtract exponents. $${2}^{3-3}=1$$ Simplify. $${2}^{0}=1$$

We see $$\frac{{a}^{m}}{{a}^{n}}$$ simplifies to a $${a}^{0}$$ and to $$1$$. So $${a}^{0}=1$$.

Definition: 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. $$\phantom{\rule{0.2em}{0ex}}{12}^{0}$$
2. $$\phantom{\rule{0.2em}{0ex}}{y}^{0}$$

Solution

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

 $${12}^{0}$$ Use the definition of the zero exponent. 1
 $${y}^{0}$$ Use the definition of the zero exponent. 1

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.

 $${\left(2x\right)}^{0}$$ Use the Product to a Power Rule. $${2}^{0}{x}^{0}$$ Use the Zero Exponent Property. $$1\cdot 1$$ Simplify. 1

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

Example

Simplify: $${\left(7z\right)}^{0}.$$

Solution

 $${\left(7z\right)}^{0}$$ Use the definition of the zero exponent. 1

Example

Simplify:

1. $$\phantom{\rule{0.2em}{0ex}}{\left(-3{x}^{2}y\right)}^{0}$$
2. $$\phantom{\rule{0.2em}{0ex}}-3{x}^{2}{y}^{0}$$

Solution

 The product is raised to the zero power. $${\left(-3{x}^{2}y\right)}^{0}$$ Use the definition of the zero exponent. $$1$$
 Notice that only the variable $$y$$ is being raised to the zero power. $${-3{x}^{2}y}^{0}$$ Use the definition of the zero exponent. $$-3{x}^{2}\cdot 1$$ Simplify. $$-3{x}^{2}$$

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