Explain the zero exponent rule.

The zero exponent rule states that any non-zero number raised to the power of zero equals one.

To understand why this is the case, let's delve into the properties of exponents. When you multiply numbers with the same base, you add their exponents. For example, \(a^m \times a^n = a^{m+n}\). Now, consider what happens when you divide numbers with the same base: \(a^m \div a^n = a^{m-n}\). If \(m\) and \(n\) are equal, this simplifies to \(a^m \div a^m = a^{m-m} = a^0\). Since any number divided by itself is 1 (as long as the number is not zero), we get \(a^0 = 1\).

Let's look at a specific example to make this clearer. Take \(2^3\). This equals 8. Now, if we divide \(2^3\) by \(2^3\), we get \(2^3 \div 2^3 = 2^{3-3} = 2^0\). Since \(2^3 \div 2^3 = 8 \div 8 = 1\), it follows that \(2^0 = 1\).

This rule applies to any non-zero number. Whether it's \(5^0\), \((-3)^0\), or even \((\frac{1}{2})^0\), the result is always 1. However, it's important to note that \(0^0\) is a special case and is generally considered undefined in mathematics.

Understanding the zero exponent rule is crucial as it simplifies many algebraic expressions and helps in solving equations efficiently. Remember, any non-zero number to the power of zero is always 1!