Simplify each exponential expression in Exercises 23–64. (−3x^2 y^5)^2

Key Concepts

Exponential Rules

Exponential rules are fundamental principles that govern the manipulation of expressions involving exponents. Key rules include the product of powers, power of a power, and power of a product. For example, when raising a power to another power, you multiply the exponents. Understanding these rules is essential for simplifying expressions like (−3x^2 y^5)^2.

Distributive Property

The distributive property is a key algebraic principle that allows you to multiply a single term by each term within a parenthesis. In the context of exponents, this means applying the exponent to each factor inside the parentheses. For instance, in (−3x^2 y^5)^2, you would distribute the exponent 2 to −3, x^2, and y^5, leading to a simplified expression.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. While this concept is not directly applicable in the given expression, understanding it is crucial for broader exponential simplifications. For example, a term like x^(-n) can be rewritten as 1/(x^n), which is important when dealing with more complex expressions involving negative bases or exponents.