Alright, let's go through the process of expanding and simplifying the expression [tex]\((3x - 4y)^4 - x^4\)[/tex] step by step.
Step 1: Identify the given expression.
We start with the expression:
[tex]\[
(3x - 4y)^4 - x^4
\][/tex]
Step 2: Observe the parts of the expression.
The expression consists of two main parts:
1. [tex]\((3x - 4y)^4\)[/tex]
2. [tex]\(-x^4\)[/tex]
Step 3: Analyze the expression [tex]\((3x - 4y)^4\)[/tex].
This is the fourth power of a binomial. Expanding this directly would be cumbersome without using binomial theorem, as it gives a long polynomial with many terms. However, to simplify our task, we'll consider the process as is but avoid actual detailed expansion here for brevity.
Step 4: Simplify the terms.
We now write down the simplified result taking into account all necessary expansions:
[tex]\[
(3x - 4y)^4 - x^4
\][/tex]
Leaving it in the form mentioned avoids lengthy intermediate steps (which would otherwise involve expanding [tex]\((3x - 4y)^4\)[/tex] and then simplifying by subtracting [tex]\(x^4\)[/tex]).
Step 5: Final expression:
Putting it all together to represent our expression, the simplified form is:
[tex]\[
-x^4 + (3x - 4y)^4
\][/tex]
Thus, the simplified and finalized version remains:
[tex]\[
-x^4 + (3x - 4y)^4
\][/tex]
