Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. -(2x⁰y⁴)³

Verified step by step guidance

1. Recognize that any variable or number raised to the zero power equals 1, so simplify \(x^0\) to 1 inside the parentheses. This changes the expression to \(-(2 \cdot 1 \cdot y^4)^3\).
2. Simplify inside the parentheses to get \(-(2y^4)^3\).
3. Apply the exponent of 3 to both the coefficient 2 and the variable term \(y^4\) inside the parentheses using the power of a product rule: \((ab)^n = a^n b^n\). This gives \(-(2^3)(y^4)^3\).
4. Use the power of a power rule for the variable term: \((y^4)^3 = y^{4 \times 3} = y^{12}\).
5. Combine the results to write the expression as \(- (2^3) y^{12}\), which simplifies to \(-8 y^{12}\).

Key Concepts

Exponents and Powers

Exponents indicate how many times a base is multiplied by itself. For example, x^3 means x multiplied three times. Understanding how to apply powers to variables and constants is essential for simplifying expressions involving exponents.

Zero Exponent Rule

Any nonzero number raised to the zero power equals 1. For instance, x^0 = 1 for x ≠ 0. This rule simplifies expressions by eliminating variables raised to the zero power, reducing complexity.

Power of a Product Rule

When raising a product to a power, apply the exponent to each factor inside the parentheses separately. For example, (ab)^n = a^n * b^n. This rule helps in distributing exponents correctly across terms in an expression.