### Part A: Determining the Length of Each Side of the Square
We are given that the area of a square is [tex]\(16x^2 - 8x + 1\)[/tex] square units. Since we know that the area of a square is the side length squared, we can find the side length by factoring the given expression completely.
Given expression:
[tex]\[
16x^2 - 8x + 1
\][/tex]
To factor this quadratic expression, we look for two binomials that multiply together to give us the original quadratic expression.
We notice that:
[tex]\[
16x^2 - 8x + 1 = (4x - 1)(4x - 1) = (4x - 1)^2
\][/tex]
Thus, the factored form of [tex]\(16x^2 - 8x + 1\)[/tex] is:
[tex]\[
(4x - 1)^2
\][/tex]
This means that the side length of the square is:
[tex]\[
4x - 1
\][/tex]
### Part B: Determining the Dimensions of the Rectangle
We are given that the area of a rectangle is [tex]\(81x^2 - 4y^2\)[/tex] square units. To find the dimensions of the rectangle, we need to factor the given expression using the difference of squares method.
Given expression:
[tex]\[
81x^2 - 4y^2
\][/tex]
The difference of squares formula tells us that [tex]\(a^2 - b^2\)[/tex] can be factored into [tex]\((a - b)(a + b)\)[/tex].
In our case:
[tex]\[
81x^2 - 4y^2 = (9x)^2 - (2y)^2
\][/tex]
Applying the difference of squares formula:
[tex]\[
81x^2 - 4y^2 = (9x - 2y)(9x + 2y)
\][/tex]
Thus, the factored form of [tex]\(81x^2 - 4y^2\)[/tex] is:
[tex]\[
(9x - 2y)(9x + 2y)
\][/tex]
This means the dimensions of the rectangle are:
[tex]\[
9x - 2y \quad \text{and} \quad 9x + 2y
\][/tex]
### Final Solution
Part A: The side length of the square is [tex]\(4x - 1\)[/tex].
Part B: The dimensions of the rectangle are [tex]\(9x - 2y\)[/tex] and [tex]\(9x + 2y\)[/tex].
