Answer :
To determine which pair of expressions can be the dimensions of the rectangle whose area is given by [tex]\(24 x^5 y^{15}\)[/tex], we need to check each pair of expressions by multiplying them and seeing if their product matches [tex]\(24 x^5 y^{15}\)[/tex]. Let's go through each pair step-by-step:
1. First pair: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
[tex]\[
(2 x^5 y^8) \times (12 x y^7) = 2 \times 12 \times x^5 \times x \times y^8 \times y^7 = 24 x^{5+1} y^{8+7} = 24 x^6 y^{15}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
2. Second pair: [tex]\(6 x^2 y^3\)[/tex] and [tex]\(4 x^3 y^5\)[/tex]
[tex]\[
(6 x^2 y^3) \times (4 x^3 y^5) = 6 \times 4 \times x^2 \times x^3 \times y^3 \times y^5 = 24 x^{2+3} y^{3+5} = 24 x^5 y^8
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
3. Third pair: [tex]\(10 x^6 y^{15}\)[/tex] and [tex]\(14 x^6 y^{15}\)[/tex]
[tex]\[
(10 x^6 y^{15}) \times (14 x^6 y^{15}) = 10 \times 14 \times x^6 \times x^6 \times y^{15} \times y^{15} = 140 x^{6+6} y^{15+15} = 140 x^{12} y^{30}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
4. Fourth pair: [tex]\(9 x^4 y^{11}\)[/tex] and [tex]\(12 x^2 y^4\)[/tex]
[tex]\[
(9 x^4 y^{11}) \times (12 x^2 y^4) = 9 \times 12 \times x^4 \times x^2 \times y^{11} \times y^4 = 108 x^{4+2} y^{11+4} = 108 x^6 y^{15}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
After checking all the pairs, none of the given dimensions multiply to [tex]\(24 x^5 y^{15}\)[/tex]. Therefore, none of the options provided are correct for matching the given area.
1. First pair: [tex]\(2 x^5 y^8\)[/tex] and [tex]\(12 x y^7\)[/tex]
[tex]\[
(2 x^5 y^8) \times (12 x y^7) = 2 \times 12 \times x^5 \times x \times y^8 \times y^7 = 24 x^{5+1} y^{8+7} = 24 x^6 y^{15}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
2. Second pair: [tex]\(6 x^2 y^3\)[/tex] and [tex]\(4 x^3 y^5\)[/tex]
[tex]\[
(6 x^2 y^3) \times (4 x^3 y^5) = 6 \times 4 \times x^2 \times x^3 \times y^3 \times y^5 = 24 x^{2+3} y^{3+5} = 24 x^5 y^8
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
3. Third pair: [tex]\(10 x^6 y^{15}\)[/tex] and [tex]\(14 x^6 y^{15}\)[/tex]
[tex]\[
(10 x^6 y^{15}) \times (14 x^6 y^{15}) = 10 \times 14 \times x^6 \times x^6 \times y^{15} \times y^{15} = 140 x^{6+6} y^{15+15} = 140 x^{12} y^{30}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
4. Fourth pair: [tex]\(9 x^4 y^{11}\)[/tex] and [tex]\(12 x^2 y^4\)[/tex]
[tex]\[
(9 x^4 y^{11}) \times (12 x^2 y^4) = 9 \times 12 \times x^4 \times x^2 \times y^{11} \times y^4 = 108 x^{4+2} y^{11+4} = 108 x^6 y^{15}
\][/tex]
This does not match [tex]\(24 x^5 y^{15}\)[/tex], so this pair is incorrect.
After checking all the pairs, none of the given dimensions multiply to [tex]\(24 x^5 y^{15}\)[/tex]. Therefore, none of the options provided are correct for matching the given area.
