College

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]

Answer :

To solve this problem, we need to determine which pair of expressions could be the dimensions (length and width) of a rectangle with the area given by the expression [tex]\( 24x^6y^{15} \)[/tex].

The area of a rectangle is found by multiplying the length by the width. Let's evaluate each option given, by multiplying the two dimensions and comparing the result with [tex]\( 24x^6y^{15} \)[/tex].

### Option 1: [tex]\( 2x^5y^8 \)[/tex] and [tex]\( 12xy^7 \)[/tex]
Multiply the two expressions:

[tex]\[
2x^5y^8 \times 12xy^7 = (2 \times 12)(x^5 \times x)(y^8 \times y^7)
= 24x^{5+1}y^{8+7}
= 24x^6y^{15}
\][/tex]

This matches the given area of [tex]\( 24x^6y^{15} \)[/tex].

### Option 2: [tex]\( 6x^2y^3 \)[/tex] and [tex]\( 4x^3y^5 \)[/tex]
Multiply the two expressions:

[tex]\[
6x^2y^3 \times 4x^3y^5 = (6 \times 4)(x^2 \times x^3)(y^3 \times y^5)
= 24x^{2+3}y^{3+5}
= 24x^5y^8
\][/tex]

This does not match the given area.

### Option 3: [tex]\( 10x^6y^{15} \)[/tex] and [tex]\( 14x^6y^{15} \)[/tex]
Multiply the two expressions:

[tex]\[
10x^6y^{15} \times 14x^6y^{15} = (10 \times 14)(x^{6+6})(y^{15+15})
= 140x^{12}y^{30}
\][/tex]

This does not match the given area.

### Option 4: [tex]\( 9x^4y^{11} \)[/tex] and [tex]\( 12x^2y^4 \)[/tex]
Multiply the two expressions:

[tex]\[
9x^4y^{11} \times 12x^2y^4 = (9 \times 12)(x^{4+2})(y^{11+4})
= 108x^6y^{15}
\][/tex]

This does not match the given area.

### Conclusion
The correct answer is the first option: [tex]\( 2x^5y^8 \)[/tex] and [tex]\( 12xy^7 \)[/tex], since their product equals the given area [tex]\( 24x^6y^{15} \)[/tex].