Simplify each expression. Assume all variables represent nonzero real numbers. -(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.
2. Rewrite the expression inside the parentheses as \(2 \cdot 1 \cdot y^{4}\), which simplifies to \$2y^{4}$.
3. Apply the exponent outside the parentheses to each factor inside: raise 2 to the 3rd power and \(y^{4}\) to the 3rd power, using the power of a product and power of a power rules.
4. Use the power of a power rule: \((y^{4})^{3} = y^{4 \times 3} = y^{12}\), and calculate \$2^{3}$ as part of the expression.
5. Don't forget the negative sign outside the parentheses; multiply it by the result of the exponentiation to write the fully simplified expression.

Key Concepts

Exponents and Powers

Exponents indicate how many times a base is multiplied by itself. For example, x^3 means x × x × x. 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. This rule simplifies terms like 2x^0y^4 by reducing x^0 to 1, making the expression easier to handle.

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 expanding and simplifying expressions like (2x^0y^4)^3.