Answer :
To find the dimensions of a rectangle from the given area expression [tex]\(24x^6y^{15}\)[/tex], we have a few options to consider, and we're looking for two expressions that multiply to give [tex]\(24x^6y^{15}\)[/tex].
Let's go through each option one by one:
1. Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply these:
[tex]\[
(2x^5y^8) \times (12xy^7) = 2 \times 12 \times x^5 \times x \times y^8 \times y^7
\][/tex]
- Calculate:
[tex]\[
= 24x^{5+1}y^{8+7} = 24x^6y^{15}
\][/tex]
- This combination matches the given area [tex]\(24x^6y^{15}\)[/tex].
2. Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply these:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = 6 \times 4 \times x^2 \times x^3 \times y^3 \times y^5
\][/tex]
- Calculate:
[tex]\[
= 24x^{2+3}y^{3+5} = 24x^5y^8
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
3. Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply these:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = 10 \times 14 \times x^{6+6} \times y^{15+15}
\][/tex]
- Calculate:
[tex]\[
= 140x^{12}y^{30}
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
4. Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply these:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = 9 \times 12 \times x^{4+2} \times y^{11+4}
\][/tex]
- Calculate:
[tex]\[
= 108x^6y^{15}
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
In conclusion, the correct dimensions of the rectangle that result in the area [tex]\(24x^6y^{15}\)[/tex] are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
Let's go through each option one by one:
1. Option 1: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply these:
[tex]\[
(2x^5y^8) \times (12xy^7) = 2 \times 12 \times x^5 \times x \times y^8 \times y^7
\][/tex]
- Calculate:
[tex]\[
= 24x^{5+1}y^{8+7} = 24x^6y^{15}
\][/tex]
- This combination matches the given area [tex]\(24x^6y^{15}\)[/tex].
2. Option 2: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply these:
[tex]\[
(6x^2y^3) \times (4x^3y^5) = 6 \times 4 \times x^2 \times x^3 \times y^3 \times y^5
\][/tex]
- Calculate:
[tex]\[
= 24x^{2+3}y^{3+5} = 24x^5y^8
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
3. Option 3: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply these:
[tex]\[
(10x^6y^{15}) \times (14x^6y^{15}) = 10 \times 14 \times x^{6+6} \times y^{15+15}
\][/tex]
- Calculate:
[tex]\[
= 140x^{12}y^{30}
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
4. Option 4: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply these:
[tex]\[
(9x^4y^{11}) \times (12x^2y^4) = 9 \times 12 \times x^{4+2} \times y^{11+4}
\][/tex]
- Calculate:
[tex]\[
= 108x^6y^{15}
\][/tex]
- This does not match the required area of [tex]\(24x^6y^{15}\)[/tex].
In conclusion, the correct dimensions of the rectangle that result in the area [tex]\(24x^6y^{15}\)[/tex] are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
