Simplify each exponential expression in Exercises 23–64. (4x^3)^−2

Verified step by step guidance

1. Start by recalling the rule of exponents: \((a^m)^n = a^{m \cdot n}\). This rule will help simplify the expression \((4x^3)^{-2}\).
2. Apply the rule of exponents to distribute the \(-2\) exponent to both the base \(4\) and \(x^3\). This gives \(4^{-2} \cdot (x^3)^{-2}\).
3. Simplify \(4^{-2}\) using the property \(a^{-n} = \frac{1}{a^n}\). This results in \(\frac{1}{4^2}\).
4. Simplify \((x^3)^{-2}\) using the same property \(a^{-n} = \frac{1}{a^n}\). This results in \(\frac{1}{x^{3 \cdot 2}}\), which simplifies further to \(\frac{1}{x^6}\).
5. Combine the simplified terms \(\frac{1}{4^2}\) and \(\frac{1}{x^6}\) into a single fraction: \(\frac{1}{4^2 \cdot x^6}\). Finally, simplify \(4^2\) to \(16\), resulting in \(\frac{1}{16x^6}\).

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of expressions involving exponents. Key rules include the product of powers, power of a power, and negative exponents. For instance, a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent, which is crucial for simplifying expressions like (4x^3)^{-2}.

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the positive exponent. For example, a term like a^{-n} can be rewritten as 1/a^n. This concept is essential for simplifying expressions with negative exponents, as it allows us to convert them into a more manageable form, facilitating further simplification.

Power of a Product

The power of a product rule states that when raising a product to an exponent, you can distribute the exponent to each factor in the product. For example, (ab)^n = a^n * b^n. This rule is particularly useful in simplifying expressions like (4x^3)^{-2}, as it allows us to separately handle the constant and the variable components.