Zero exponent rule

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