What is the value of \( a^0 \) when \( a \neq 0 \)?

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.