Gavin is subdividing land into two plots, where one plot is in the shape of a square and the other plot is in the shape of a rectangle. The square plot of land has a side length of 6x^3 feet, and the rectangular plot of land has a length of 3x^5 feet and a width of 7x^2 feet. Use the properties of exponents to determine the expression that represents the area for each plot of land. Then analyze which plot of land has a larger area if x is 3.
For the square plot of land, the area is \(s^2\), and the side lengths are \(6x^3\), so \(6x^3 \cdot 6x^3 = 36x^6\).
For the rectangular plot of land, the area is \(l \cdot w\), and the length and width are \(3x^5\) and \(7x^2\), so \(3x^5 \cdot 7x^2 = 21x^7\).
If \(x\) is 3, then we put 3 for x.
\(36 \cdot 3^6 = 36 \cdot 729 = 26244\)sq ft
and
\(21 \cdot 3^7 = 21 \cdot 2187 = 45927\)sq ft.
So, the rectangular plot of land has a larger area.
