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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].
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].
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