Write each expression without negative exponents, and evaluate if possible. Assume all variables represent nonzero real numbers. See Example 4. (-4)^-3

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 the negative exponent rule to the expression \((-4)^{-3}\), rewriting it as \(\frac{1}{(-4)^3}\).
3. Calculate the denominator by raising \(-4\) to the power of 3: \((-4)^3 = (-4) \times (-4) \times (-4)\).
4. Multiply the factors in the denominator step-by-step: first \((-4) \times (-4)\), then multiply the result by \((-4)\) again.
5. Write the final expression as a fraction with numerator 1 and the calculated denominator, completing the 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 the base to the denominator.

Evaluating Powers of Negative Numbers

When raising a negative number to a power, consider whether the exponent is even or odd. An odd exponent preserves the negative sign, while an even exponent results in a positive value. For example, (-4)^3 = -64 because 3 is odd.

Simplifying Expressions with Exponents

Simplifying expressions involves applying exponent rules systematically, such as converting negative exponents to positive, and then calculating numerical values when possible. This process ensures expressions are in standard form and easier to interpret.