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

Verified step by step guidance

1. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a \neq 0\). This means that any expression with a negative exponent can be rewritten as the reciprocal with a positive exponent.
2. Apply this rule to the given expression \((4x)^{-2}\). Rewrite it as \(\frac{1}{(4x)^2}\).
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\( to get \)16\(, so the expression becomes \)\frac{1}{16x^2}$.
5. Since the problem states to evaluate if possible and all 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/a^n, where a ≠ 0. This rule allows rewriting expressions without negative exponents by moving factors between numerator and denominator.

Exponentiation of a Product

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For instance, (ab)^n = a^n * b^n. This property helps simplify expressions like (4x)^-2 by applying 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 operations like taking reciprocals or raising to negative powers are valid.