To determine the simplified expression, we must consider the given options:
A) [tex]\(\frac{16 a^2}{b^2}\)[/tex]
B) [tex]\(16 a^2\)[/tex]
C) [tex]\(\frac{4 a^2}{b^2}\)[/tex]
D) [tex]\(\frac{4 b^2}{a^2}\)[/tex]
To answer this question, we recognize that the most simplified form is the one that can't be further reduced while maintaining the same variables and structure.
Analyzing each option:
- Option A, [tex]\(\frac{16 a^2}{b^2}\)[/tex], is a rational expression and seems already simplified because [tex]\(16 a^2\)[/tex] and [tex]\( b^2\)[/tex] have no common factors that can be canceled out.
- Option B, [tex]\(16 a^2\)[/tex], represents a different structure as it misses the division by [tex]\(b^2\)[/tex], and thus might not reflect the same original structure needing to be simplified.
- Option C, [tex]\(\frac{4 a^2}{b^2}\)[/tex], suggests that the numerator has been simplified to [tex]\(4 a^2\)[/tex], but if there was no context suggesting any steps were missed to reach [tex]\(16 a^2\)[/tex], it may not be fully correct.
- Option D, [tex]\(\frac{4 b^2}{a^2}\)[/tex], flips both numerators and denominators which represent a different structure, intuitively not being directly simplified.
After logical consideration of each of these given options recognizable from mathematical simplifications:
The correct simplified version that fits the given conditions and remains true to the expected straightforward rational simplification of the target expression is:
[tex]\[ \boxed{\frac{16 a^2}{b^2}} \][/tex]
Thus, the simplified expression is [tex]\(\frac{16 a^2}{b^2}\)[/tex], making Option A the correct answer.
