The “ Zero Power Rule” Explained

The “ Zero Power Rule” Explained

Exponents seem pretty straightforward, right? Raise a number to the power of 1 means you have one of that number, raise to the power of 2 means you have two of the number multiplied together, power 3 means three of the number multiplied together and so on.

But what about the zero power? Why is any non-zero number raised to the power of zero equal 1? And what happens when we raise zero to the zero power? Is it still 1?

Watch the video or read below to find out!

Click here to subscribe to Math Hacks

Warm-Up Example

Let?s begin by examining division of values with exponents.

Image for post

Recall exponents represent repeated multiplication. So we can rewrite the above expression as:

Image for post

Since 2/2 = 1, cancel out three sets of 2/2. This leaves 2 ? 2, or 2 squared.

Image for post

Of course we can take a shortcut and subtract the number of 2?s on bottom from the number of 2?s on top. Since these quantities are represented by their respective exponents, all we need to do is write the common base with the difference in exponent values as the power.

Image for post

If we generalize this rule, we have the following where n represents a non-zero real number and x and y are also real numbers.

Image for postRule for Dividing Numbers with a Common Base

Exploring the Zero Power

From here it is easy to derive the explanation for why any non-zero number raised to the zero power equals 1. Again, let?s look at a concrete example.

We know that any non-zero number divided by itself equals 1. So I can write the following:

Image for post

This is the same as writing:

Image for post

Now I?ll utilize the exponent rule from above to rewrite the left hand side of this equation.

Image for post

Of course, this is equivalent to:

Image for post

We can use the same process as in this example, along with the generalized rule above, to show that any non-zero real number raised to the zero power must result in 1.

What About Zero to the Zero Power?

This is where things get tricky. The above method breaks because, of course, dividing by zero is a no-no. Let?s examine why.

We?ll begin with looking at a common divide by zero ERROR.

How about 20? Let?s look at why we can?t do this.

Image for post

Division is really just a form of multiplication, so what happens if I rewrite the above equation as:

Image for post

What value could possible satisfy this equation for x?

There is no value! Any number times zero results in zero, it can never equal 2. Therefore, we say division by zero is undefined. There is no possible solution.

Now let?s look at 00.

Image for post

Again, rewrite it as a multiplication problem.

Image for post

Here we encounter a very different situation. The solution for x could be ANY real number! There is no way to determine what x is. Hence, 0/0 is considered indeterminate*, not undefined.

If we try to use the above method with zero as the base to determine what zero to the zero power would be, we come to halt immediately and cannot continue because we know that 00 ? 1, but is indeterminate.

So what does zero to the zero power equal?

This is highly debated. Some believe it should be defined as 1 while others think it is 0, and some believe it is undefined. There are good mathematical arguments for each, and perhaps it is most correctly considered indeterminate.

Despite this, the mathematical community is in favor of defining zero to the zero power as 1, at least for most purposes.

Perhaps a helpful definition of exponents for the amateur mathematician is as follows:

Image for post

By including the ?1? in the definition, we can conclude that any number (including zero) repeated zero times results in 1.

*Calculus Note:

The notion of indeterminate forms is commonplace in Calculus. A simple example of why 0/0 is indeterminate can be found by examining some basic limits.

Image for postImage for postFollow Math Hacks on Instagram

These limits cannot be directly evaluated since they are indeterminate forms. Instead we must use L?hopital?s rule, taking the derivative of the numerator and denominator separately, to find the solutions are 2 and 3 respectively.

When dealing with an equation that results in an indeterminate form of zero to the zero power while practicing Calculus, make sure to implement techniques for indeterminates, such as L?hopital?s rule, to correctly evaluate the limit.

? STAY CONNECTED ?

Stay up-to-date with everything Math Hacks is up to!

Instagram | Facebook | Twitter

Next Lesson: Common Misconception About Probability

Subscribe Now! Math Hacks is on YouTube!

Join me as we tackle math together one problem at a time. Spreading math love + self-empowerment. Subscribe for new?

www.youtube.com

Related Reads

Top 10 Secrets of Pascal?s Triangle

Binomial Theorem, Fibonacci Sequence, Sierpinski Triangle & More

medium.com

How To ?Complete the Square ? Visually

and the Trouble with Memorization

medium.com

Using Factor Trees to Find GCFs and LCMs

We?ve been discussing the Euclidean Algorithm and how to find the greatest common divisor of two numbers. The algorithm?

medium.com

Guide To Fractions in 10 Simple Facts

lesson twenty-two

medium.com

2

No Responses

Write a response