Answer :
To solve the problem of finding the dimensions of a rectangle with an area of [tex]\(24x^6y^{15}\)[/tex], we need to check each set of given dimensions to see which pair, when multiplied, equals [tex]\(24x^6y^{15}\)[/tex].
Let's evaluate each option:
1. Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the lengths:
[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 product matches the expression for the area, so these could be the dimensions of the rectangle.
2. Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the lengths:
[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]
- The product here does not match the given area.
3. Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the lengths:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = (10 \times 14)(x^6 \times x^6)(y^{15} \times y^{15}) = 140x^{12}y^{30}
\][/tex]
- This product does not match the given area.
4. Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the lengths:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = (9 \times 12)(x^4 \times x^2)(y^{11} \times y^4) = 108x^{6}y^{15}
\][/tex]
- This product does not match the given area.
After evaluating all the options, the dimensions [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex] provide the correct product that equals the area of [tex]\(24x^6y^{15}\)[/tex].
Let's evaluate each option:
1. Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the lengths:
[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 product matches the expression for the area, so these could be the dimensions of the rectangle.
2. Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the lengths:
[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]
- The product here does not match the given area.
3. Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the lengths:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = (10 \times 14)(x^6 \times x^6)(y^{15} \times y^{15}) = 140x^{12}y^{30}
\][/tex]
- This product does not match the given area.
4. Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the lengths:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = (9 \times 12)(x^4 \times x^2)(y^{11} \times y^4) = 108x^{6}y^{15}
\][/tex]
- This product does not match the given area.
After evaluating all the options, the dimensions [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex] provide the correct product that equals the area of [tex]\(24x^6y^{15}\)[/tex].
