Certainly! Let's start by analyzing the problem given.
We need to find side length of the square if the area of the square is represented by [tex]\( A \)[/tex].
The options provided for the area of the square in the given problem are:
1. [tex]\( 3 q^2 r^6\left|s^3\right| \)[/tex]
2. [tex]\( 3 q^2 r^4\left|s^3\right| \)[/tex]
3. [tex]\( 18 q^8 r^{16} s^{12} \)[/tex]
4. [tex]\( 81 q^8 r^{16} s^{12} \)[/tex]
And we need to find side length of the square whose area is equal to these expressions.
1. For the first option, [tex]\( A = 3 q^2 r^6\left|s^3\right| \)[/tex] units:
The area [tex]\(A\)[/tex] of a square is given by [tex]\( A = \text{side}^2 \)[/tex].
So, for [tex]\[ \text{side} = \sqrt{3 q^2 r^6\left|s^3\right|} \][/tex]
To simplify this:
[tex]\[ \text{side} = \sqrt{3} \sqrt{q^2} \sqrt{r^6} \sqrt{\left|s^3\right|} \][/tex]
[tex]\[ \text{side} = \sqrt{3} \cdot q \cdot r^3 \cdot \left|s\right|^\frac{3}{2} \][/tex]
2. For the second option, [tex]\( A = 3 q^2 r^4\left|s^3\right| \)[/tex] units:
[tex]\[ \text{side} = \sqrt{3 q^2 r^4\left|s^3\right|} \][/tex]
To simplify this:
[tex]\[ \text{side} = \sqrt{3} \sqrt{q^2} \sqrt{r^4} \sqrt{\left|s^3\right|} \][/tex]
[tex]\[ \text{side} = \sqrt{3} \cdot q \cdot r^2 \cdot \left|s\right|^\frac{3}{2} \][/tex]
3. For the third option, [tex]\( A = 18 q^8 r^{16} s^{12} \)[/tex] units:
[tex]\[ \text{side} = \sqrt{18 q^8 r^{16} s^{12}} \][/tex]
To simplify this:
[tex]\[ \text{side} = \sqrt{18} \sqrt{q^8} \sqrt{r^{16}} \sqrt{s^{12}} \][/tex]
[tex]\[ \text{side} = \sqrt{18} \cdot q^4 \cdot r^8 \cdot s^6 \][/tex]
4. For the fourth option, [tex]\( A = 81 q^8 r^{16} s^{12} \)[/tex] units:
[tex]\[ \text{side} = \sqrt{81 q^8 r^{16} s^{12}} \][/tex]
To simplify this:
[tex]\[ \text{side} = \sqrt{81} \sqrt{q^8} \sqrt{r^{16}} \sqrt{s^{12}} \][/tex]
[tex]\[ \text{side} = 9 \cdot q^4 \cdot r^8 \cdot s^6 \][/tex]
Thus, the side lengths for each square given the area are:
1. [tex]\( \sqrt{3} \cdot q \cdot r^3 \cdot \left|s\right|^\frac{3}{2}\)[/tex]
2. [tex]\( \sqrt{3} \cdot q \cdot r^2 \cdot \left|s\right|^\frac{3}{2}\)[/tex]
3. [tex]\( \sqrt{18} \cdot q^4 \cdot r^8 \cdot s^6\)[/tex]
4. [tex]\( 9 \cdot q^4 \cdot r^8 \cdot s^6\)[/tex]
To answer the question, for the area to be [tex]\( \left(18 q^8 r^{16} s^{12} \right) \)[/tex] square units, the side length should be [tex]\( \sqrt{18} q^4 r^8 s^6 \)[/tex]. This confirms that our initial area analysis is consistent with our question's expectations. So, the correct answer is detailed in steps and consistent with the given areas and expected side lengths.
