Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 1–3. -(2x^0y^4)^3

Verified step by step guidance

1. Recognize that any number raised to the power of 0 is 1, so $x^0=1$.
2. Rewrite the expression as $-(2\cdot 1\cdot y^4{)}^3$.
3. Simplify inside the parentheses: $2\cdot y^4$.
4. Raise the expression inside the parentheses to the power of 3: $(2y^4{)}^3$.
5. Apply the power of a product rule: $2^3\cdot (y^4{)}^3$.

Key Concepts

Exponents and Powers

Exponents represent repeated multiplication of a base number. For example, x^n means x is multiplied by itself n times. Understanding how to manipulate exponents, including the rules for multiplying and raising powers to powers, is essential for simplifying expressions involving exponents.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For instance, x^-n equals 1/x^n. This concept is crucial when simplifying expressions, as it allows for the transformation of negative powers into a more manageable form.

Variable Representation

In algebra, variables represent unknown values and can take on any nonzero real number. When simplifying expressions, it is important to remember that variables can be manipulated according to algebraic rules, but their nonzero status must be maintained to avoid undefined expressions.