In Exercises 85–116, simplify each exponential expression. (x⁻⁵y⁸/3)⁻⁴

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of expressions involving exponents. Key rules include the product of powers, quotient of powers, and power of a power, which dictate how to simplify expressions like (a^m * a^n = a^(m+n)) and (a^m / a^n = a^(m-n)). Understanding these rules is essential for simplifying complex exponential expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). This concept is crucial when simplifying expressions, as it allows for the transformation of negative exponents into a more manageable form, facilitating further simplification of the overall expression.

Distributive Property of Exponents

The distributive property of exponents states that when raising a product to a power, each factor in the product is raised to that power. For instance, (ab)^n = a^n * b^n. This property is vital for simplifying expressions like (x^m * y^n)^p, as it allows for the distribution of the exponent across the terms, leading to a clearer and more simplified expression.