Let's simplify and combine the given mathematical expressions step-by-step.
### Step 1: Analyze the First Expression
The first part of the question is:
[tex]\[ 16a^4 + 4a^2b^2 + 1 \][/tex]
This is a polynomial expression involving terms with variables [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
### Step 2: Analyze the Second Expression
The next term is:
[tex]\[ b^2 \][/tex]
This is simply a squared term involving the variable [tex]\(b\)[/tex].
### Step 3: Analyze the Third Expression
The third part is a fractional expression:
[tex]\[ \frac{54x^3}{81x^4 + 9x^2 + 1} \][/tex]
This is a rational function involving the variable [tex]\(x\)[/tex].
### Step 4: Combine the Expressions
Now let's combine all three parts together. We will write them in the order given and ensure that each term is properly accounted for.
Combining the polynomial, the squared term, and the rational function, we get:
[tex]\[ 16a^4 + 4a^2b^2 + b^2 + \frac{54x^3}{81x^4 + 9x^2 + 1} + 1 \][/tex]
So, the final combined expression is:
[tex]\[ 16a^4 + 4a^2b^2 + b^2 + \frac{54x^3}{81x^4 + 9x^2 + 1} + 1 \][/tex]
This is the simplified and completely combined form of the given expressions.
