Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. (4x)^-2

Verified step by step guidance

1. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a\) is a nonzero number and \(n\) is a positive integer.
2. Apply this rule to the expression \((4x)^{-2}\). Rewrite it as \(\frac{1}{(4x)^2}\) to eliminate the negative exponent.
3. Next, simplify the denominator by applying the exponent to both the coefficient and the variable inside the parentheses: \((4x)^2 = 4^2 \cdot x^2\).
4. Calculate \$4^2\( as \)16\(, so the expression becomes \)\frac{1}{16x^2}$.
5. Since the problem states to evaluate if possible and variables represent nonzero real numbers, this is the simplified expression without negative exponents.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻ⁿ = 1/aⁿ, where a ≠ 0. This rule allows rewriting expressions without negative exponents by moving factors between numerator and denominator.

Exponent Rules for Products

When raising a product to a power, apply the exponent to each factor inside the parentheses. For instance, (ab)ⁿ = aⁿbⁿ. This helps simplify expressions like (4x)⁻² by distributing the exponent to both 4 and x.

Evaluating Expressions with Variables

When variables represent nonzero real numbers, expressions can be simplified by applying algebraic rules without concern for division by zero. This assumption ensures that rewriting expressions with negative exponents is valid and that evaluation is possible if numerical values are given.