Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. -5^-4

Verified step by step guidance

1. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a \neq 0\) and \(n\) is a positive integer.
2. Apply this rule to the expression \(-5^{-4}\). Notice that the negative sign is separate from the base with the exponent, so the expression is equivalent to \(-(5^{-4})\).
3. Rewrite \$5^{-4}\( using the negative exponent rule: \)5^{-4} = \frac{1}{5^4}$.
4. Substitute back into the expression: \(-5^{-4} = -\frac{1}{5^4}\).
5. Evaluate \$5^4\( if needed by multiplying \)5 \times 5 \times 5 \times 5\(, then write the final expression as \)-\frac{1}{5^4}$ (or the evaluated denominator) 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 the base to the denominator.

Order of Operations and Exponentiation

Exponentiation is performed before multiplication and addition. In the expression -5^-4, the exponent applies only to 5, not the negative sign. This means you first calculate 5^-4, then apply the negative sign, ensuring correct evaluation.

Evaluating Powers of Numbers

Evaluating powers involves multiplying the base by itself the number of times indicated by the exponent. For negative exponents, after rewriting as a reciprocal, compute the positive power. For example, 5^-4 = 1/5^4 = 1/625.