What Is 10 Times 0

10 to the Power of 0: the Zero Exponent Rule and the Power of Zero Explained

Exponents are important in the fiscal world, in scientific notation, and in the fields of epidemiology and public health. Then what are they, and how do they work?

Exponents are written like \(3^2\) or \(10^iii\).

Simply what happens when y'all raise a number to the \(0\) power like this?

\[10^0 = \text{?}\]

This article will go over

the basics of exponents,
what they mean, and
information technology will bear witness that \(x^0\) equals \(1\) using negative exponents

All I'm assuming is that you take an understanding of multiplication and division.

Exponents are made up of a base and exponent (or power)

Offset, let's start with the parts of an exponent.

There are two parts to an exponent:

the base
the exponent or power

At the first, nosotros had an exponent \(three^ii\). The "3" here is the base, while the "ii" is the exponent or power.

We read this equally

Iii is raised to the ability of two.

Three to the power of 2.

More than by and large, exponents are written every bit \(a^b\), where \(a\) and \(b\) can be any pair of numbers.

Exponents are multiplication for the "lazy"

Now that nosotros have some understanding of how to talk nigh exponents, how do we find what number it equals?

Using our case from above, we tin write out and expand "three to the power of ii" as

\[3^2 = 3 \times 3 = nine\]

The left-well-nigh number in the exponent is the number we are multiplying over and once again. That is why you are seeing multiple iii'due south. The correct-almost number in the exponent is the number of multiplications we do. So for our case, the number 3 (the base) is multiplied two times (the exponent).

Some more examples of exponents are:

\[10^three = 10 \times ten \times x = yard\]

\[2^{10} = 2 \times 2 \times 2 \times 2 \times ii \times two \times ii \times 2 \times 2 \times 2 = 1024 \]

More generally, we can write these exponents as

\[\textcolor{orange}{b}^\textcolor{blue}{due north} = \underbrace{\textcolor{orangish}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blue}{n} \textrm{ times}}\]

where, the \(\textcolor{orange}{\text{letter ``b'' is the base of operations}}\) nosotros are multiplying over and over once more and the \(\textcolor{bluish}{\text{letter ``n'' is power}}\) or \(\textcolor{blueish}{\text{exponent}}\), which is the number of times we are multiplying the base by itself.

For these examples above, the exponent values are relatively small. But you lot can imagine if the powers are very large, it becomes redundant to keep writing the numbers over and over again using multiplication signs.

In sum, exponents assistance make writing these long multiplications more than efficient.

Numbers to the power of zero are equal to one

The previous examples show powers of greater than one, but what happens when information technology is zero?

The quick answer is that whatsoever number, \(b\), to the power of zero is equal to one.

\[b^0 = 1\]

Based on our previous definitions, we just need null of the base value. Here, let's accept our base of operations number be 10.

\[x^0 = ? = i\]

Just what does a "zero" number of base numbers mean? Why does this happen?

We tin can figure this out by dividing multiple times to subtract the power value until we go to zero.

Allow's beginning with

\[10^iii = 10 \times 10 \times 10 = yard\]

To decrease the powers, nosotros need to briefly understand the concepts of

combining exponents
powers of one

In our quest to decrease the exponent from \(ten^three\) ("x to the third power") to \(10^0\) ("ten to the zeroth power"), we will keep on doing the contrary of multiplying, which is dividing.

\[\frac{10^3}{10} = \frac{10 \times 10 \times 10}{10} = \frac{chiliad}{ten} = 100\]

The right-most parts of this will probably make sense. Simply how practice we write exponents when we have \(ten^3\) divided by \(10\)?

How powers of ane piece of work

Get-go, whatsoever \(\textcolor{orangish}{\text{exponents with powers of one}}\) are equal to but \(\textcolor{bluish}{\text{the base number}}\).

\[\textcolor{orangish}{b^1} = \textcolor{blueish}{b}\]

There is only one value being "multiplied" so we are getting the value itself.

Nosotros need this "power of one" definition so we tin rewrite the fraction with exponents.

\[\frac{10^3}{10} = \frac{ten^iii}{ten^1}\]

How to subtract exponents to zero

As a reminder, 1 manner to figure out how \(10^0\) is equal to 1 is to keep on dividing by x until we get to an exponent of nothing.

We know from the right side of the equation above we should get 100 from \(\frac{ten^3}{x^i}\).

\[ \frac{10^3}{ten} = \frac{10^iii}{10^i} = \frac{10 \times 10 \times 10}{10^i} \]

Before nosotros stop dividing past one 10, we can multiply the top and bottom by one as placeholders when we abolish numbers out.

\[ \frac{ten \times 10 \times x}{x^ane} = \frac{10 \times 10 \times x \times 1}{10^1 \times one} = \frac{10 \times ten \times \abolish{10} \times 1}{\cancel{x^i} \times 1} = \frac{10 \times ten \times ane}{ane}\]

From this, we can see we become 100 once again.

\[ \frac{10 \times x \times 1}{1} = \frac{10 \times x}{i} = \frac{10^2}{1} = \frac{100}{1} \]

We can divide past x two more times to finally get to \(ten^0\).

\[ \frac{10^2 \times one}{10 \times ten \times one} = \frac{\cancel{x} \times \abolish{x} \times i}{\cancel{10} \times \cancel{x} \times one} = \frac{10^0 \times 1}{ane} = \frac{one}{1} = 1 \]

Considering we divided past 2 10'southward when we simply had two x'southward in the superlative of the fraction, we accept zero tens in the pinnacle. Having aught tens pretty much means we get \(10^0\).

How negative exponents work

At present, the \(x^0\) kind of comes out of nowhere, so permit'south explore this some more using "negative exponents".

More than more often than not, this repetitive dividing by the same base of operations is the same as multiplying past "negative exponents".

A negative exponent is a manner to rewrite segmentation.

\[ \frac{1}{\textcolor{purple}{b^northward}}= \textcolor{green}{b^{-n}}\]

A \(\textcolor{green}{\text{negative exponent}}\) can be re-written every bit a fraction with the denominator (or the bottom of a fraction) with the \(\textcolor{purple}{\text{same exponent only with a positive power}}\) (the left side of this equation).

Now, using negative exponents, we can bear witness the previous division in another fashion.

\[ \frac{x^2 \times 1}{10 \times x \times ane} = \frac{10^2}{10^ii} = 10^2 \times \frac{ane}{10^2} = 10^two \times 10^{-2} \]

Note, i rule of exponents is that when you lot multiply exponents with the same base of operations number (remember, our base number here is ten), you can add together the exponents.

\[ 10^2 \times ten^{-2} = 10^{ii + (-two)} = 10^{2 - 2} = ten^{0} \]

Putting it together

Knowing this, we tin combine each of these equations above to summarize our issue.

\[ \textcolor{purple}{\frac{ten^ii}{10^2}} = x^2 \times ten^{-2} = 10^{ii + (-2)} = 10^{2 - ii} = \textcolor{blue}{10^{0}} \textcolor{orange}{= 1} \]

We know that \(\textcolor{purple}{\text{dividing a number by itself}}\) will \(\textcolor{orange}{\text{equal to ane}}\). And we've shown that \(\textcolor{purple}{\text{dividing a number by itself}}\) as well equals \(\textcolor{blue}{\text{x to the zilch power}}\). Math says that things that are equal to the same affair are also equal to each other.

Thus, \(\textcolor{blue}{\text{ten to the cypher power}}\) is \(\textcolor{orangish}{\text{equal to one}}\). This exercise above generalizes to whatever base number, so any number to the power of naught is equal to one.

In summary

Exponents are convenient ways to practice repetitive multiplication.

By and large, exponents follow this pattern below, with some \(\textcolor{orange}{\text{base number}}\) being multiplied over and once more \(\textcolor{blue}{\text{``n'' number of times}}\).

\[\textcolor{orangish}{b}^\textcolor{blue}{northward} = \underbrace{\textcolor{orange}{b} \times \dots \times \textcolor{orange}{b}}_{\textcolor{blue}{northward} \textrm{ times}}\]

Using negative exponents, nosotros can have what we know from multiplication and division (like for the fraction 10 over 10,\(\frac{x}{10}\)) to show that \(b^0\) is equal to one for any number \(b\) (like \(x^0 = i\)).

Follow me on Twitter and check out my personal web log where I share some other insights and helpful resources for programming, statistics, and machine learning.

Thank you for reading!

Larn to code for gratis. freeCodeCamp'due south open source curriculum has helped more than forty,000 people get jobs as developers. Go started