In this chapter we will learn about value of numbers with power 0 with solved examples.

At the end of the post we will also look at the proof of the zero exponent rule.

To understand this chapter, you should have** basic idea of exponents**. Click the red link to read about the topic in detail.

## What is Zero exponent rule ?

According to the rule, **any number with power 0 equals to 1**.

If a is any given number, then;

\mathtt{( a)^{0} \ \Longrightarrow \ 1}

### Is this rule applicable for large numbers ?

Yes !!

No matter how big the number is. If it has 0 exponent, its final value will be 1.

**For example;**

\mathtt{( 1297)^{0} =1\ }\\\ \\ \mathtt{( 905671)^{0} =1}

### Does 0 exponent rule work for fractions ?

Yes !!

The value of any fraction raised to the power 0 is 1.

i.e. \mathtt{\left(\frac{a}{b}\right)^{0} =1} **For example;**

\mathtt{\left(\frac{6}{5}\right)^{0} =1\ }\\\ \\ \mathtt{\left(\frac{895}{121}\right)^{0} =1}

### Negative number with power 0

The zero exponent rule is also **applicable for negative number**.

Hence, any negative number with exponent 0 measures 1.

For example;

\mathtt{( -5)^{0} =1\ }\\\ \\ \mathtt{( -21)^{0} =1}

### Zero exponent in variable

We know that variable can have different values and are generally represented by English alphabet “y”.

But if variable is raised to the power zero,** the final result will be 1.**

\mathtt{( y)^{0} =1}

### Zero exponent on math expression

Zero exponent rule also **works in math expression.**

Hence, if any given expression is raised to the power 0, its value will be 1.

This means that in this case, you won’t have to solve the further expression.

**For example, consider the below case.**

\mathtt{( 2x+3y-7)^{0} =1\ }

### Zero exponent rule for number 0

This case is **not properly defined by mathematics expert.**

Some say the value is 0 while other claim it to be 1.

Hence for \mathtt{( 0)^{0}} , there is no consensus among the experts.

## Proof of zero exponent rule

To prove the zero exponent rule, let’s start with the product rule.**According to the product rule of exponents;**

\mathtt{x^{m+n} =x^{m} .x^{n}}

Let m = 1 and n = 0.

Putting the values;

\mathtt{x^{1+0} =x^{1} .x^{0}}\\\ \\ \mathtt{x^{1} =x^{1} .x^{0}}

Divide both the sides by \mathtt{x^{1}}\\\ \\

\mathtt{\frac{\mathtt{x^{1}}}{x^{1}} =\frac{\mathtt{x^{1} .x^{0}}}{x^{1}}}\\\ \\ \mathtt{1\ =\ x^{0}} **Hence Proved.****Conclusion** : Any number (except 0) raised to the power 0 values 1