The area of a rectangle, [tex]A = l \cdot w[/tex], is represented by the expression [tex]24 x^6 y^{15}[/tex]. Which could be the dimensions of the rectangle?

A. [tex]2 x^5 y^8[/tex] and [tex]12 x y^7[/tex]

B. [tex]6 x^2 y^3[/tex] and [tex]4 x^3 y^5[/tex]

C. [tex]10 x^6 y^{15}[/tex] and [tex]14 x^6 y^{15}[/tex]

D. [tex]9 x^4 y^{11}[/tex] and [tex]12 x^2 y^4[/tex]

To find the dimensions of the rectangle that result in the area [tex]\( 24x^6y^{15} \)[/tex], we check each option by multiplying the dimensions provided and seeing if they give the correct area.

1. Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the dimensions:
[tex]\[
(2x^5y^8) \times (12xy^7) = 2 \times 12 \times x^{5+1} \times y^{8+7}
\][/tex]
[tex]\[
= 24x^6y^{15}
\][/tex]
- The product is [tex]\(24x^6y^{15}\)[/tex], which matches the given area.

2. Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the dimensions:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = 6 \times 4 \times x^{2+3} \times y^{3+5}
\][/tex]
[tex]\[
= 24x^5y^8
\][/tex]
- The product is [tex]\(24x^5y^8\)[/tex], which does not match the given area.

3. Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the dimensions:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = 10 \times 14 \times x^{6+6} \times y^{15+15}
\][/tex]
[tex]\[
= 140x^{12}y^{30}
\][/tex]
- The product is [tex]\(140x^{12}y^{30}\)[/tex], which does not match the given area.

4. Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the dimensions:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = 9 \times 12 \times x^{4+2} \times y^{11+4}
\][/tex]
[tex]\[
= 108x^6y^{15}
\][/tex]
- The product is [tex]\(108x^6y^{15}\)[/tex], which does not match the given area.

The dimensions that correctly form the area [tex]\(24x^6y^{15}\)[/tex] are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex] (Option 1).