Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3.(-6x²)³

Verified step by step guidance

1. Identify the expression to simplify: \((-6x^2)^3\).
2. Apply the power of a product rule: \((ab)^n = a^n \cdot b^n\).
3. Separate the expression into two parts: \((-6)^3\) and \((x^2)^3\).
4. Calculate \((-6)^3\) using the rule for powers of negative numbers.
5. Apply the power of a power rule to \((x^2)^3\): \((x^m)^n = x^{m \cdot n}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. For example, in the expression a^n, 'a' is the base and 'n' is the exponent. Understanding how to apply exponent rules, such as multiplying powers or raising a power to a power, is essential for simplifying expressions.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For instance, a^(-n) equals 1/(a^n). While this concept is not directly applicable in the given expression, recognizing how negative exponents work is crucial for simplifying expressions that may involve them in other contexts.

Polynomial Expansion

Polynomial expansion involves multiplying out expressions that contain variables raised to powers. In the case of (-6x²)³, this requires applying the power of a product rule, which states that (ab)^n = a^n * b^n. This means that both the coefficient and the variable must be raised to the power of three, leading to a simplified polynomial expression.