Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. x³y⁵- ( ——— )⁰ z

Verified step by step guidance

1. Recognize that any nonzero number or expression raised to the power of zero is equal to 1.
2. Identify the expression inside the parentheses: \( \frac{x^3y^5}{z} \).
3. Apply the zero exponent rule: \( \left( \frac{x^3y^5}{z} \right)^0 = 1 \).
4. Understand that the expression simplifies directly to 1, regardless of the values of \(x\), \(y\), and \(z\), as long as they are nonzero.
5. Conclude that the simplified form of the given expression is 1.

Key Concepts

Exponents and Powers

Exponents represent repeated multiplication of a base number. For example, x³ means x multiplied by itself three times. Understanding the rules of exponents, such as the power of a product and the power of a quotient, is essential for simplifying expressions involving variables raised to powers.

Zero Exponent Rule

The zero exponent rule states that any nonzero number raised to the power of zero equals one. This means that (a)⁰ = 1 for any nonzero a. This rule is crucial for simplifying expressions where a term is raised to the zero power, as it effectively eliminates that term from the expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying algebraic rules to reduce the expression to its simplest form. This process often includes using the properties of exponents, such as the zero exponent rule, to eliminate terms and clarify the expression, making it easier to work with in further calculations.