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Chapter 4: Problem 38

\((-4)^{-3}\)

Short Answer

Expert verified
-\( \frac{1}{64}\)

Step by step solution

01

Understand the Negative Exponent

A negative exponent means that we take the reciprocal of the base and then apply the positive exponent. For instance, \((-4)^{-3} = \frac{1}{(-4)^3}\).
02

Apply the Positive Exponent

Now calculate \((-4)^3\). This means multiplying -4 by itself three times: \((-4) \times (-4) \times (-4) = -64\).
03

Combine the Results

Substitute the result from Step 2 back into the equation from Step 1: \((-4)^{-3} = \frac{1}{-64} = -\frac{1}{64}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
When dealing with negative exponents, the concept of a reciprocal is crucial. A reciprocal is simply the inverse of a number. For instance, the reciprocal of 4 is \(\frac{1}{4}\). This means if you have \(-4^{-3}\), you need to find the reciprocal of \(-4\). This process helps in transforming a negative exponent into a positive one. Here, \(-4^{-3}\) becomes \(\frac{1}{-4^3}\). It's a straightforward operation that makes handling negative exponents much easier. By converting the exponent from negative to positive, your calculations become more manageable.
Positive Exponent
Once the exponent is made positive by taking the reciprocal, you need to understand how to deal with positive exponents. The rule for positive exponents is to multiply the base by itself as many times as indicated by the exponent. In the case of \(-4^3\), it means you multiply -4 by itself three times:\(-4 \times -4 \times -4 = -64\). Positive exponents indicate repeated multiplication. Each multiplication should be handled carefully, keeping track of negative signs to ensure accuracy. Remember: multiplying an odd number of negative numbers results in a negative number, while multiplying an even number results in a positive number.
Multiplication of Negative Numbers
Understanding the rules of multiplying negative numbers is key to solving problems involving negative exponents. When you multiply two negative numbers, the result is positive. This is because a negative times a negative equals a positive. However, if you have an odd number of negative factors, the result will be negative. For instance, in \(-4^{-3}\), once we transform the expression into \(\frac{1}{-4^3}\) and calculate \(-4 \times -4 \times -4\), the product is -64, because there are three (an odd number) of negative multiplications. This step-by-step approach ensures accuracy and a deeper understanding of the underlying principles.

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