Answer :
To find the dimensions of the rectangle whose area is given by [tex]\(24x^6y^{15}\)[/tex], we need to identify two expressions that multiply together to produce this area.
Let's evaluate each option one by one:
1. Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
To check if these dimensions are correct, multiply the two expressions:
[tex]\[
(2x^5y^8) \times (12xy^7) = (2 \times 12) \times (x^5 \times x^1) \times (y^8 \times y^7)
\][/tex]
Simplifying, we get:
[tex]\[
24x^{5+1}y^{8+7} = 24x^6y^{15}
\][/tex]
This matches the given area. Therefore, these dimensions are correct.
2. Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
Multiply these terms:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = (6 \times 4) \times (x^{2+3}) \times (y^{3+5})
\][/tex]
Simplifying, we get:
[tex]\[
24x^5y^8
\][/tex]
This doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
3. Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
Multiply these terms:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = (10 \times 14) \times (x^{6+6}) \times (y^{15+15})
\][/tex]
Simplifying, we get:
[tex]\[
140x^{12}y^{30}
\][/tex]
This doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
4. Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
Multiply these terms:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = (9 \times 12) \times (x^{4+2}) \times (y^{11+4})
\][/tex]
Simplifying, we get:
[tex]\[
108x^6y^{15}
\][/tex]
This also doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
Therefore, the correct dimensions of the rectangle are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex], as they multiply to give the area [tex]\(24x^6y^{15}\)[/tex].
Let's evaluate each option one by one:
1. Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
To check if these dimensions are correct, multiply the two expressions:
[tex]\[
(2x^5y^8) \times (12xy^7) = (2 \times 12) \times (x^5 \times x^1) \times (y^8 \times y^7)
\][/tex]
Simplifying, we get:
[tex]\[
24x^{5+1}y^{8+7} = 24x^6y^{15}
\][/tex]
This matches the given area. Therefore, these dimensions are correct.
2. Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
Multiply these terms:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = (6 \times 4) \times (x^{2+3}) \times (y^{3+5})
\][/tex]
Simplifying, we get:
[tex]\[
24x^5y^8
\][/tex]
This doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
3. Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
Multiply these terms:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = (10 \times 14) \times (x^{6+6}) \times (y^{15+15})
\][/tex]
Simplifying, we get:
[tex]\[
140x^{12}y^{30}
\][/tex]
This doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
4. Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
Multiply these terms:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = (9 \times 12) \times (x^{4+2}) \times (y^{11+4})
\][/tex]
Simplifying, we get:
[tex]\[
108x^6y^{15}
\][/tex]
This also doesn't match the given area [tex]\(24x^6y^{15}\)[/tex].
Therefore, the correct dimensions of the rectangle are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex], as they multiply to give the area [tex]\(24x^6y^{15}\)[/tex].
