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# Zero Exponents

An exponential number is a function that is expressed in the form of ${x}^{n}$ , where x represents a constant, known as the base; and n, the exponent of the function, can be any number. The exponent defines how many times the base is multiplied by itself. For example, ${3}^{4}$ represents the operation $3×3×3×3=81$ .

Many students find it strange that anything raised to the power of 0 is 1. A lot of students think it should be 0, just like anything multiplied by 0 is 0. But remember, you are not multiplying by the exponent, but using the exponent to tell how many times to multiply a number by itself.

## Using the product of powers property to see how zero exponents work

We can use the product of powers property to see exactly how the 0 exponent works.

${9}^{0}×{9}^{1}={9}^{\left(0+1\right)}={9}^{1}$

We already know that ${9}^{1}=9$ . The product of powers property says that ${9}^{0}×9=9$ . If we try to multiply $0×9$ , we get 0, so ${9}^{0}$ can't equal 0. We do know that $1×9=9$ , so ${9}^{0}$ has to equal 1.

The general rule is that for all real numbers x where $x\ne 0$ , we have:

${x}^{0}=1$

## Using the quotient of powers property to show how zero exponents work

We can also use the quotient of powers property to show that a number with a 0 exponent equals 1.

We know that $\frac{{x}^{2}}{{x}^{2}}=1$ using the quotient of powers property because when we divide numbers with the same base, we subtract the exponents.

That means that $\frac{{x}^{2}}{{x}^{2}}={x}^{\left(2-2\right)}={x}^{0}$ , and we already know that $\frac{{x}^{2}}{{x}^{2}}=1$ , so we can see that ${x}^{0}=1$ . That is, any number (except 0) raised to the power of 0 equals 1.

Note that ${0}^{0}$ is undefined.

## Get help learning about zero exponents

Tutoring is an excellent way for your student to gain a more thorough understanding of zero exponents. Face it, zero exponents can be a little bit tricky. They are not intuitive. If your student could use some help with them, a tutor is just the right choice. A tutor can meet with your student in 1-on-1 sessions using customized learning materials that focus on their most effective way of learning. A tutor can also try multiple ways of looking at zero exponents until they find one that works well for your student. To get connected with an expert tutor who can help your student understand zero exponents and more math topics, contact the Educational Directors at Varsity Tutors today.

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