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

Verified step by step guidance

1. Recall the zero exponent rule: for any nonzero expression \(a\), \(a^0 = 1\). This means that any expression raised to the power of zero simplifies to 1.
2. Identify the expression inside the parentheses: \(\left( \frac{x^3 y^5}{z} \right)^0\). Since the entire fraction is raised to the zero power, the value of this expression is 1 regardless of the values of \(x\), \(y\), and \(z\) (as long as they are nonzero).
3. Rewrite the original expression using this simplification: \(x^3 y^5 - 1\).
4. Since the problem asks to simplify the expression, note that no further simplification is possible because \(x^3 y^5\) and 1 are unlike terms and cannot be combined.
5. Therefore, the simplified form of the expression is \(x^3 y^5 - 1\).

Key Concepts

Zero Exponent Rule

Any nonzero base raised to the zero power equals 1. This means that for any expression (A)⁰, where A ≠ 0, the value simplifies directly to 1 regardless of A's complexity.

Properties of Exponents

Exponents indicate repeated multiplication. When simplifying expressions with exponents, rules such as product, quotient, and power of a power help combine or reduce terms efficiently.

Simplification of Algebraic Expressions

Simplifying involves reducing expressions to their simplest form by applying arithmetic and algebraic rules, including combining like terms and applying exponent rules, to make expressions easier to interpret or solve.