Exponent of 0

How do you simplify a power with an exponent of 0?

We know that it's pretty easy to simplify positive exponents. For example, \(3^{5}\) just means that you multiply 3 by itself 5 times.

\(3^{5}= 3\times 3\times 3\times 3\times 3 = 243\)

\(3^{0}= ?\)

But what do you do if the exponent is a 0? How do you multiply 3 by itself zero times? Does it equal 3? Does it equal 0?

Nope! It actually equals 1.

In fact, ANY number raised to an exponent of 0 will equal 1. 

\(x^{0}= 1\)

Examples

\(3^{0}= 1\)

\(15^{0}= 1\)

\(987^{0}= 1\)

Why does it equal 1?

When I first learned how to simplify an exponent of 0, I was really confused. I wasn't sure how to multiply a number by itself 0 times, but I was pretty sure that 1 was NOT a logical answer. 

Then I saw the following explanation and it suddenly made sense.

To see why any number with an exponent of 0 equals 1, let's start by listing and simplifying the powers of 3...

\(3^{5}= 3\times 3\times 3\times 3\times 3 = 243\)

\(3^{4}= 3\times 3\times 3\times 3 = 81\)

\(3^{3}= 3\times 3\times 3 = 27\)

\(3^{2}= 3\times 3 = 9\)

\(3^{1}= 3\)

\(3^{0}=\) ?

Take a look at the pattern.

What do you notice?

What happens to the answers as the exponents get smaller?

When I look at it, I notice that it's a geometric sequence.

Every time the exponent decreases by 1, the simplified answer is one-third of the previous term's answer.

If we continue this pattern of dividing each term by 3 to find the next term, it suddenly becomes clear why \(3^{0}\) equals 1.

The term right before \(3^{0}\) is \(3^{1}\)(which equals 3). If you divide it by 3 to find one-third of it, you get 1. So, \(3^{0}=1\).

If you continue this pattern, you'll see why negative exponents create fractions. 


So, what about other numbers?

What if you need to calculate \(4^{0}\) or \(27^{0}\) or any other number with an exponent of 0?

Well, let’s try it out with the powers of 4…

\(4^{4}= 4\times 4\times 4\times 4 =  \color{black}{256}\)

\(4^{3}= 4\times 4\times 4 =  \color{black}{64}\)

\(4^{2}= 4\times 4 =  \color{black}{16}\)

\(4^{1}=  \color{black}{4}\)

\(4^{0}=\) ?

Notice the pattern...

\(256\div4=64\)

\(64\div4=16\)

\(16\div4=4\)

\(4\div4=\) ?


In this pattern, each term is one-fourth of the previous term. And when you divide 4 by 4, that equals 1. So, \(4^{0} = 1\). 

Next, let's try the powers of 27...

\(27^{5}= \color{black}{14,348,907}\)

\(27^{4}=\color{black}{531,441}\)

\(27^{3}= \color{black}{19,683}\)

\(27^{2}= \color{black}{729}\)

\(27^{1}=  \color{black}{27}\)

\(27^{0}=\) ?

Notice the pattern...

\(14,348,907\div27=531,441\)

\(531,441\div27=19,683\)

\(19,683\div27=729\)

\(729\div27=27\)

\(27\div27=\) ?


In this pattern, each term is one twenty-seventh of the previous term. And when you divide 27 by 27, you can see that \(27^{0} = 1\). 

Fun Fact:

Some mathematicians argue that \(0^{0}\) does NOT equal 1.

It is "undefined" like \(\frac{0}{0}\).

Any other number with an exponent of 0 will have a similar pattern.

So, any power with an exponent of 0 will equal 1, regardless of what number is in the base


Negative Numbers with an Exponent of 0

If you have a negative number with an exponent of 0, you have to pay attention to the parentheses. 

\((-12)^{0}=1\)

In this example, the negative sign is applied to the 12.

Then, the -12 is raised to an exponent of 0.

So, \((-12)^{0}\) equals positive 1. 

\(-12^{0}=-1\)

In this example, 12 is the base of the power and it's raised to an exponent of 0, which results in a positive 1. 

Then, the negative sign is applied to the 1.

So, \(-12^{0}\) equals negative 1. 

If the negative sign is outside of the parentheses, it's NOT part of the power and it needs to be applied AFTER the power is simplified.

If the negative sign is inside of the parentheses, then it IS part of the base of the power and it will equal a positive number when it's raised to an even exponent

Constants

The constant in a polynomial is the term whose variable has an exponent of 0. A variable with an exponent of 0 can be simplified to equal 1. When you multiply that 1 by the coefficient of the term, you get a constant.

\(3x^2-5x^{1}+17x^0\)

The exponent of 0 can be simplified like this...\(17x^0=17(1)=17\).

The exponent of 1 can be simplified like this...\(-5x^1=-5x\).

So, the simplified polynomial is...

\(3x^2-5x+17\)

Resources

Free Online Practice Problems

Khan Academy - Exponents (Basic)

Khan Academy - Exponents

Free Printable Worksheets

Homeschool Math - Whole Number Bases

Homeschool Math - Fraction & Decimal Bases

The worksheets from Homeschool Math will generate new numbers every time you refresh the page. Answer keys are included.

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