What is the value of $$ a^0 $$ when $$ a
eq 0 $$?

Question

What is the value of $$ a^0 $$ when $$ a 
eq 0 $$?

UpStudy AI · Accepted Answer

For any non-zero real number $$ a $$, the expression $$ a^0 $$ is equal to **1**. This might initially seem counterintuitive, but it is consistent with the rules of exponents. Here's why:

### Understanding Exponent Rules

One of the fundamental properties of exponents is:

$$
a^m \div a^n = a^{m-n}
$$

Let's apply this property to understand $$ a^0 $$:

$$
a^n \div a^n = a^{n-n} = a^0
$$

But $$ a^n \div a^n $$ is also equal to 1 (since any non-zero number divided by itself is 1):

$$
a^n \div a^n = 1
$$

Therefore:

$$
a^0 = 1
$$

### Example

Consider $$ a = 5 $$:

$$
5^0 = 1
$$

Similarly, for $$ a = -3 $$:

$$
(-3)^0 = 1
$$

### Important Note

- **Undefined Expression:** The expression $$ 0^0 $$ is **undefined** in mathematics because it can lead to contradictory results depending on the context. However, as long as $$ a \neq 0 $$, $$ a^0 $$ is always 1.

### Summary

$$
a^0 = 1 \quad \text{for any} \quad a \neq 0
$$

This rule is widely accepted and used consistently across different areas of mathematics, including algebra, calculus, and beyond.
$$ a^0 = 1 $$ when $$ a \neq 0 $$.

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