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Answer :
To solve this problem, we need to determine which pair of dimensions, when multiplied together, equals the given area of the rectangle, [tex]\(24x^6y^{15}\)[/tex].
Let's examine each possible pair of dimensions and see if they match the given area.
1. First Pair: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
To find the area, multiply these two expressions:
[tex]\[
(2x^5y^8) \times (12xy^7) = 2 \times 12 \times x^{5+1} \times y^{8+7} = 24x^6y^{15}
\][/tex]
This matches the given area, so this pair could be the dimensions of the rectangle.
2. Second Pair: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
Multiply these two expressions:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = 6 \times 4 \times x^{2+3} \times y^{3+5} = 24x^5y^8
\][/tex]
This does not match the given area, so this pair is not correct.
3. Third Pair: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
Multiply these two expressions:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = 10 \times 14 \times x^{6+6} \times y^{15+15} = 140x^{12}y^{30}
\][/tex]
This does not match the given area, so this pair is not correct.
4. Fourth Pair: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
Multiply these two expressions:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = 9 \times 12 \times x^{4+2} \times y^{11+4} = 108x^6y^{15}
\][/tex]
This does not match the given area, so this pair is not correct.
Since only the first pair of dimensions, [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex], results in the correct area, [tex]\(24x^6y^{15}\)[/tex], this is the correct answer.
Let's examine each possible pair of dimensions and see if they match the given area.
1. First Pair: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
To find the area, multiply these two expressions:
[tex]\[
(2x^5y^8) \times (12xy^7) = 2 \times 12 \times x^{5+1} \times y^{8+7} = 24x^6y^{15}
\][/tex]
This matches the given area, so this pair could be the dimensions of the rectangle.
2. Second Pair: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
Multiply these two expressions:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = 6 \times 4 \times x^{2+3} \times y^{3+5} = 24x^5y^8
\][/tex]
This does not match the given area, so this pair is not correct.
3. Third Pair: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
Multiply these two expressions:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = 10 \times 14 \times x^{6+6} \times y^{15+15} = 140x^{12}y^{30}
\][/tex]
This does not match the given area, so this pair is not correct.
4. Fourth Pair: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
Multiply these two expressions:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = 9 \times 12 \times x^{4+2} \times y^{11+4} = 108x^6y^{15}
\][/tex]
This does not match the given area, so this pair is not correct.
Since only the first pair of dimensions, [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex], results in the correct area, [tex]\(24x^6y^{15}\)[/tex], this is the correct answer.
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