Verified Answer :

  • Apply the power rule \((ab)^n = a^n b^n\) to rewrite \((2x)^{-4}\) as \(\frac{1}{16} x^{-4}\).
  • Multiply this result by \(4x^3\) to get \(\left(4 \cdot \frac{1}{16}\right) \cdot \left(x^3 \cdot x^{-4}\right)\).
  • Simplify the constant to \(\frac{1}{4}\) and the variable to \(x^{-1}\) or \(\frac{1}{x}\).
  • Conclude that the expression simplifies to \(\boxed{\frac{1}{4x}}\).

Explanation

  1. Analysis of the Expression

The goal is to simplify the expression \((4x^3)(2x)^{-4}\) and match it with one of the given choices. We identify the expression as a product of two terms, where the second term involves a negative exponent. We will use the rules of exponents to simplify the expression systematically. Specifically, we will use:

  1. The power of a product rule: \((ab)^n = a^n \cdot b^n\)
  2. The definition of a negative exponent: \(a^{-n} = \frac{1}{a^n}\)
  3. The product rule for exponents: \(a^m \cdot a^n = a^{m+n}\)
  1. Simplifying the Term with the Exponent

First, let's simplify the second part of the expression, \((2x)^{-4}\). Applying the power of a product rule \((ab)^n = a^n b^n\), we get:
\[(2x)^{-4} = 2^{-4} \cdot x^{-4}\]
Next, we evaluate the constant part \(2^{-4}\):
\[2^{-4} = \frac{1}{2^4} = \frac{1}{16}\]
Substituting this back, we have:
\[(2x)^{-4} = \frac{1}{16}x^{-4}\]

  1. Combining Constants and Variables

Now we substitute this back into the original expression \((4x^3)(2x)^{-4}\):
\[(4x^3) \cdot \left(\frac{1}{16}x^{-4}\right)\]
We can group the constants and the variables separately:
\[\left(4 \cdot \frac{1}{16}\right) \cdot (x^3 \cdot x^{-4})\]
Calculating the constant part:
\[4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4}\]
Calculating the variable part using the rule \(x^m \cdot x^n = x^{m+n}\):
\[x^3 \cdot x^{-4} = x^{3 + (-4)} = x^{-1} = \frac{1}{x}\]

  1. Final Result and Conclusion

Multiplying the simplified constant part and the simplified variable part, we obtain:
\[\frac{1}{4} \cdot \frac{1}{x} = \frac{1}{4x}\]
Comparing this result to the given options, we find that it matches the first option. Therefore, the equivalent expression is \(\boxed{\frac{1}{4x}}\).

Examples

Imagine you are a packaging engineer working on a production line where items are placed in boxes. You need to calculate the number of items that fit into a batch of containers, where the capacity of each container and the number of containers are expressed as algebraic variables. For instance, if you have a batch size determined by \((4x^3)\) and you are downsizing by a factor of \((2x)^4\), simplifying this algebraic expression helps you quickly determine the final number of items per container, ensuring that your production flow is accurate and efficient. Understanding how to manipulate these types of algebraic expressions is essential for scaling systems and optimizing resources in various engineering and manufacturing contexts.

Related Questions