Simplify each exponential expression: (-2x^4y^3)^3

Verified step by step guidance

1. Identify the expression to simplify: \((-2x^4y^3)^3\).
2. Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\).
3. Distribute the exponent 3 to each factor inside the parentheses: \((-2)^3, (x^4)^3, (y^3)^3\).
4. Simplify each term: \((-2)^3 = -8\), \((x^4)^3 = x^{12}\), \((y^3)^3 = y^9\).
5. Combine the simplified terms: \(-8x^{12}y^9\).

Key Concepts

Exponential Rules

Exponential rules govern how to manipulate expressions involving exponents. Key rules include the power of a product, which states that (ab)^n = a^n * b^n, and the power of a power, which states that (a^m)^n = a^(m*n). Understanding these rules is essential for simplifying expressions like (-2x^4y^3)^3.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^-n = 1/a^n. While this concept is not directly applicable in the given expression, recognizing how negative bases interact with exponents is crucial for broader exponential simplifications.

Distributive Property

The distributive property allows us to multiply a single term by two or more terms inside parentheses. In the context of exponents, this means applying the exponent to each factor within the parentheses. For the expression (-2x^4y^3)^3, this property helps in distributing the exponent across -2, x^4, and y^3 to simplify the expression effectively.