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

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Recall the rule for negative exponents: for any nonzero number \(a\) and integer \(n\), \(a^{-n} = \frac{1}{a^n}\). This means a negative exponent indicates the reciprocal of the base raised to the positive exponent.

Apply this rule to the expression \((-4)^{-3}\). Rewrite it as \(\frac{1}{(-4)^3}\) to eliminate the negative exponent.

Calculate the denominator \((-4)^3\) by multiplying \(-4\) by itself three times: \((-4) \times (-4) \times (-4)\).

Evaluate the multiplication step-by-step: first \((-4) \times (-4)\), then multiply the result by \(-4\) again.

Write the final expression as \(\frac{1}{\text{the value you found}}\). This is the expression without negative exponents and evaluated.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) = 1/(a^n). This rule allows rewriting expressions without negative exponents by moving the base to the denominator.

Evaluating Powers of Numbers

Evaluating powers involves multiplying the base by itself as many times as indicated by the exponent. For negative bases raised to an integer exponent, consider the sign carefully: an odd exponent keeps the negative sign, while an even exponent results in a positive value.

Assumption of Nonzero Variables

Assuming variables are nonzero ensures that expressions with variables in denominators are defined and division by zero is avoided. This assumption is crucial when rewriting negative exponents as reciprocals to maintain valid expressions.

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