Question

the area of a square piece of land in square units can be represented by the expression ( 9q^{4}8r^{8}s^{6} ). what is the length of one side of the piece of land?
options:
( 3q^{2}r^{6}s^{3} ) units
( 3q^{2}r^{4}s^{3} ) units
( 18q^{8}r^{16}s^{12} ) units
( 81q^{8}r^{16}s^{12} ) units

Explanation:

Step1: Recall the area of a square formula

The area \( A \) of a square is given by \( A = s^2 \), where \( s \) is the length of a side. So, to find the side length, we need to take the square root of the area. The area here is \( 9q^{4}8r^{6}s^{6} \) (assuming the expression is \( 9q^{4}8r^{6}s^{6} \), maybe a typo and should be \( 9q^{4}r^{8}s^{6} \)? Wait, looking at the options, let's assume the area is \( 9q^{8}r^{6}s^{6} \) (maybe miswritten). Wait, the options have \( 3q^{2}r^{4}s^{3} \)? Wait, let's re-express. Let's suppose the area is \( 9q^{4}r^{8}s^{6} \)? No, the options: first option is \( 3q^{2}r^{4}s^{3} \)? Wait, the area is a square, so side is square root of area. Let's take the area as \( 9q^{4}r^{8}s^{6} \) (maybe the original is \( 9q^{4}r^{8}s^{6} \)). Then square root of 9 is 3, square root of \( q^{4} \) is \( q^{2} \), square root of \( r^{8} \) is \( r^{4} \), square root of \( s^{6} \) is \( s^{3} \). Wait, but the first option is \( 3q^{2}r^{4}s^{3} \)? Wait, maybe the area is \( 9q^{4}r^{8}s^{6} \), then square root is \( 3q^{2}r^{4}s^{3} \). Wait, let's check the options. The first option is \( 3q^{2}r^{4}s^{3} \)? Wait, the user's image: the area is "the expression \( 9q^{4}8r^{6}s^{6} \)"? Maybe a typo, should be \( 9q^{8}r^{6}s^{6} \) or \( 9q^{4}r^{8}s^{6} \). Alternatively, maybe the area is \( 9q^{4}r^{8}s^{6} \), then side is \( \sqrt{9q^{4}r^{8}s^{6}} = 3q^{2}r^{4}s^{3} \), which matches the first option? Wait, no, the first option is \( 3q^{2}r^{4}s^{3} \)? Wait, the options are:

1. \( 3q^{2}r^{4}s^{3} \) units (maybe)
2. \( 3q^{2}r^{4}s^{3} \) units (duplicate?)

Wait, maybe the area is \( 9q^{4}r^{8}s^{6} \), so square root: \( \sqrt{9} = 3 \), \( \sqrt{q^{4}} = q^{2} \), \( \sqrt{r^{8}} = r^{4} \), \( \sqrt{s^{6}} = s^{3} \), so side is \( 3q^{2}r^{4}s^{3} \), which is the first option.

Step2: Calculate the square root of the area

Given area \( A = 9q^{4}r^{8}s^{6} \) (assuming correction), then side \( s = \sqrt{A} = \sqrt{9q^{4}r^{8}s^{6}} \).

Using the property \( \sqrt{ab} = \sqrt{a}\sqrt{b} \) and \( \sqrt{x^n} = x^{\frac{n}{2}} \):

• \( \sqrt{9} = 3 \)
• \( \sqrt{q^{4}} = q^{\frac{4}{2}} = q^{2} \)
• \( \sqrt{r^{8}} = r^{\frac{8}{2}} = r^{4} \)
• \( \sqrt{s^{6}} = s^{\frac{6}{2}} = s^{3} \)

Multiplying these together: \( 3 \times q^{2} \times r^{4} \times s^{3} = 3q^{2}r^{4}s^{3} \) units.

Answer:

\( 3q^{2}r^{4}s^{3} \) units (assuming the first option is this, as per the calculation)