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

Verified step by step guidance

1. Recognize that the expression is a power of a product: \((-6x^{2})^{3}\). According to the exponent rules, \((ab)^{n} = a^{n} b^{n}\), so you can apply the exponent to both \(-6\) and \(x^{2}\) separately.
2. Apply the exponent to the constant term: \((-6)^{3}\). This means multiplying \(-6\) by itself three times.
3. Apply the exponent to the variable term: \((x^{2})^{3}\). Use the power of a power rule, which states \((x^{m})^{n} = x^{m \times n}\).
4. Combine the results from the previous two steps to write the expression as a product of the simplified constant and the simplified variable term.
5. Write the final simplified expression by multiplying the constants and expressing the variable with the new exponent.

Key Concepts

Exponentiation of a Product

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For example, (ab)^n = a^n * b^n. This rule allows us to simplify expressions like (-6x²)³ by applying the exponent to both -6 and x².

Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m*n). This means when an exponent is raised to another exponent, you multiply the exponents. In the expression (-6x²)³, the x² raised to the 3rd power becomes x^(2*3) = x^6.

Handling Negative Bases with Exponents

When raising a negative number to an exponent, the sign depends on whether the exponent is even or odd. An odd exponent preserves the negative sign, while an even exponent results in a positive value. For (-6)³, since 3 is odd, the result is negative.