gavin is subdividing land into two plots, where one is in the shape of a square and the other is in the shape of a rectangle. The square plot of land has a side length of 6x^3 feet The rectangular plot of land has a length of 3x^5 feet and a width of 7x^2 feet. 1) Use the properties of exponents to determine the expression that represents the area of each plot of land. 2) Then determine which plot of land has the larger area if x=3.
1) The area of a square is given by the formula A = side length squared. In this case, the side length of the square plot of land is 6x^3. Therefore, the area of the square plot of land is (6x^3)^2.
Using the properties of exponents, when raising a power to another power, you multiply the exponents. Therefore, (6x^3)^2 can be rewritten as 6^2 * (x^3)^2.
Simplifying further, 6^2 is equal to 36, and (x^3)^2 is equal to x^(3*2) which is x^6. Therefore, the expression that represents the area of the square plot of land is 36x^6.
The area of a rectangle is given by the formula A = length * width. In this case, the length of the rectangular plot of land is 3x^5 and the width is 7x^2. Therefore, the area of the rectangular plot of land is (3x^5)(7x^2).
Again, using the properties of exponents, when multiplying two powers with the same base, you add the exponents. Therefore, (3x^5)(7x^2) can be simplified as 3*7 * (x^5 * x^2).
Simplifying further, 3*7 is equal to 21, and (x^5 * x^2) is equal to x^(5+2) which is x^7. Therefore, the expression that represents the area of the rectangular plot of land is 21x^7.
2) To determine which plot of land has the larger area when x = 3, we need to evaluate the expressions for both the square plot of land and the rectangular plot of land.
For the square plot of land: Area = 36(3^6) = 36 * 729 = 26,244.
For the rectangular plot of land: Area = 21(3^7) = 21 * 2,187 = 45,927.
Therefore, when x = 3, the rectangular plot of land has the larger area.