# Proving a^0 = 1 for non-zero values of a | Homework Help

• thinkies
In summary, to prove that a^0 = 1 if a is not equal to 0, we can use the definition of a^x and the laws of exponents to show that setting m=0 in (a^n)*(a^m)=a^(n+m) yields a^0 = 1. This is a basic and easy question, but a precise definition of a^x is needed to prove it.
thinkies

## Homework Statement

Prove that a^0 = 1 if a is not equal to 0.

## The Attempt at a Solution

Well,since a is not equal to 0, I replace it with another number.

(1^0)^0 = 1

(6^0)^0 = 1

ETC

Is this enough to prove that a^0 = 1 if a is not equal to 0.Even thought its a basic and easy question...im doubting.

Thx

hhmmm any1?...

You haven't 'proved' anything. You just wrote down some numbers. What's your definition of a^x? If it's e^(log(a)*x) the answer is pretty easy, just take the log of both sides. To prove something you need a precise definition of the thing you are trying to prove. What does a^x mean?

Are you just assuming the laws of exponents? Like (a^n)*(a^m)=a^(n+m)? If so, then set m=0 and solve for a^m.

Dick said:
Are you just assuming the laws of exponents? Like (a^n)*(a^m)=a^(n+m)? If so, then set m=0 and solve for a^m.

Thanks, i though about doing that before, don't know what came up in mind.

## What is the definition of a^0?

The notation a^0 is defined as the exponentiation of a number a to the power of 0. This is equivalent to multiplying a by itself 0 times, which results in the value of 1.

## Why is it important to prove that a^0 = 1 if a=/=0?

Proving this mathematical property is important because it is a fundamental rule in algebra and calculus. It is also used in various mathematical and scientific calculations, such as in the simplification of equations and finding the derivative of a function.

## How can we prove that a^0 = 1 if a=/=0?

The proof relies on the definition of exponentiation and the properties of exponents. We can start by writing a^0 as a^1/a^1, and then using the rule that a^m/a^n = a^(m-n). This leads to a^1-1 = a^0. Since any number raised to the power of 0 is equal to 1, this shows that a^0 = 1.

## What happens if we use a value of 0 for a^0?

If we use a value of 0 for a^0, the result is undefined. This is because any number raised to the power of 0 is equivalent to multiplying that number by itself 0 times, which is not a well-defined operation.

## Are there any exceptions to the rule that a^0 = 1 if a=/=0?

No, there are no exceptions to this rule. It holds true for all real numbers, including negative numbers and fractions. However, the rule is not applicable for complex numbers as the concept of exponentiation is different for them.

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