Answer :
To solve this problem, we need to find which pair of rectangle dimensions, when multiplied together, gives us the given area expression [tex]\(24x^6y^{15}\)[/tex].
Let's look at the options one by one:
1. Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Calculate [tex]\(2x^5y^8 \times 12xy^7\)[/tex]:
[tex]\(2 \times 12 = 24\)[/tex]
[tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex]
[tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
- The product is [tex]\(24x^6y^{15}\)[/tex], which matches the given area.
2. Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Calculate [tex]\(6x^2y^3 \times 4x^3y^5\)[/tex]:
[tex]\(6 \times 4 = 24\)[/tex]
[tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
[tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
- The product is [tex]\(24x^5y^8\)[/tex], which does not match the given area.
3. Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Calculate [tex]\(10x^6y^{15} \times 14x^6y^{15}\)[/tex]:
[tex]\(10 \times 14 = 140\)[/tex]
[tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
[tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
- The product is [tex]\(140x^{12}y^{30}\)[/tex], which does not match the given area.
4. Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Calculate [tex]\(9x^4y^{11} \times 12x^2y^4\)[/tex]:
[tex]\(9 \times 12 = 108\)[/tex]
[tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
[tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
- The product is [tex]\(108x^6y^{15}\)[/tex], which does not match the given area.
After checking all the options, the correct dimensions for the rectangle that give the area [tex]\(24x^6y^{15}\)[/tex] are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
Let's look at the options one by one:
1. Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Calculate [tex]\(2x^5y^8 \times 12xy^7\)[/tex]:
[tex]\(2 \times 12 = 24\)[/tex]
[tex]\(x^5 \times x = x^{5+1} = x^6\)[/tex]
[tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
- The product is [tex]\(24x^6y^{15}\)[/tex], which matches the given area.
2. Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Calculate [tex]\(6x^2y^3 \times 4x^3y^5\)[/tex]:
[tex]\(6 \times 4 = 24\)[/tex]
[tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
[tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
- The product is [tex]\(24x^5y^8\)[/tex], which does not match the given area.
3. Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Calculate [tex]\(10x^6y^{15} \times 14x^6y^{15}\)[/tex]:
[tex]\(10 \times 14 = 140\)[/tex]
[tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
[tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
- The product is [tex]\(140x^{12}y^{30}\)[/tex], which does not match the given area.
4. Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Calculate [tex]\(9x^4y^{11} \times 12x^2y^4\)[/tex]:
[tex]\(9 \times 12 = 108\)[/tex]
[tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
[tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
- The product is [tex]\(108x^6y^{15}\)[/tex], which does not match the given area.
After checking all the options, the correct dimensions for the rectangle that give the area [tex]\(24x^6y^{15}\)[/tex] are [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex].
