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

Verified step by step guidance

1. Rewrite the expression using the property of negative exponents: \((a/b)^{-n} = (b/a)^{n}\). This gives us \((3/x^{-5}y^{8})^{4}\).
2. Apply the property of exponents \((a^{m})^{n} = a^{m \cdot n}\) to each part of the expression: \((3)^{4} / (x^{-5})^{4} (y^{8})^{4}\).
3. Simplify each part: \(3^{4}\) becomes \(81\), \((x^{-5})^{4}\) becomes \(x^{-20}\), and \((y^{8})^{4}\) becomes \(y^{32}\).
4. Combine the simplified parts into a single expression: \(81 / (x^{-20} y^{32})\).
5. Use the property of negative exponents \(a^{-m} = 1/a^{m}\) to rewrite \(x^{-20}\) as \(1/x^{20}\), resulting in the final expression: \(81x^{20}/y^{32}\).

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.