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Chapter 5: Problem 32

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

Short Answer

Expert verified
-64f^3

Step by step solution

01

Understand the Problem

The problem requires finding the cube of the term \((-4f)\). This means raising the entire expression to the power of 3.
02

Apply the Exponent

Recognize that \((-4f)^3\) means \((-4f) \times (-4f) \times (-4f)\).
03

Calculate the Product

Multiply the constants and the variables separately: \((-4) \times (-4) = 16\) and \[16 \times (-4) = -64\]. For the variable \((f)\), \((f \times f \times f) = f^3\).
04

Combine the Results

Combine the numeric and variable part to form the final result: \((-4f)^3 = -64f^3\).

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Key Concepts

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

Exponentiation
Exponentiation is the process of raising a number to a power. In the context of this problem, we are working with cubing, or raising a number to the power of 3. When you see an expression like \((-4f)^3\), it means you multiply \((-4f)\) by itself three times: \((-4f) \times (-4f) \times (-4f)\). This operation helps us understand how both the constants and variables interact under multiplication.
Multiplication of Constants
When multiplying constants, follow these simple steps:
  • First, multiply the constants together. For instance, in our exercise, we had \((-4) \times (-4)\).
  • Unlike addition or subtraction, two negative numbers multiplied together give a positive result, so \((-4) \times (-4) = 16\).
  • Then, we multiply this result by the remaining constant: \(16 \times (-4)\).
  • This interaction changes the sign of the product, resulting in \(-64\).
Understanding how to handle the signs and arithmetic makes the process straightforward.
Variable Exponents
When dealing with variables raised to a power, you simply multiply the variable by itself the required number of times. In our example of \((-4f) \times (-4f) \times (-4f)\), we handle the variable \(f\):
  • Each \(f\) is multiplied together: \(f \times f \times f\).
  • The result is \(f^3\).
Combined with the constant result from earlier, the final expression becomes \(-64f^3\). This systematic approach to working with exponents ensures accuracy in solving binomial cubing problems.

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