Answer :
To find which pair of dimensions can be used to form the rectangle with an area represented by the expression [tex]\(24x^6y^{15}\)[/tex], we will calculate the product of each pair of proposed dimensions and verify if it equals [tex]\(24x^6y^{15}\)[/tex].
### Given Options:
1. [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
2. [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
3. [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
4. [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
### Calculation:
#### Option 1:
- Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 12 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^5 \times x^1 = x^{5+1} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
- Result: [tex]\(24x^6y^{15}\)[/tex]
#### Option 2:
- Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the coefficients: [tex]\(6 \times 4 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
- Result: [tex]\(24x^5y^8\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
#### Option 3:
- Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the coefficients: [tex]\(10 \times 14 = 140\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
- Result: [tex]\(140x^{12}y^{30}\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
#### Option 4:
- Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the coefficients: [tex]\(9 \times 12 = 108\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
- Result: [tex]\(108x^6y^{15}\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
### Conclusion:
Of the given options, only Option 1 (with dimensions [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]) correctly multiplies to the given area [tex]\(24x^6y^{15}\)[/tex]. Thus, these are the possible dimensions of the rectangle.
### Given Options:
1. [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
2. [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
3. [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
4. [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
### Calculation:
#### Option 1:
- Dimensions: [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]
- Multiply the coefficients: [tex]\(2 \times 12 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^5 \times x^1 = x^{5+1} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^8 \times y^7 = y^{8+7} = y^{15}\)[/tex]
- Result: [tex]\(24x^6y^{15}\)[/tex]
#### Option 2:
- Dimensions: [tex]\(6x^2y^3\)[/tex] and [tex]\(4x^3y^5\)[/tex]
- Multiply the coefficients: [tex]\(6 \times 4 = 24\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^2 \times x^3 = x^{2+3} = x^5\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^3 \times y^5 = y^{3+5} = y^8\)[/tex]
- Result: [tex]\(24x^5y^8\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
#### Option 3:
- Dimensions: [tex]\(10x^6y^{15}\)[/tex] and [tex]\(14x^6y^{15}\)[/tex]
- Multiply the coefficients: [tex]\(10 \times 14 = 140\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^6 \times x^6 = x^{6+6} = x^{12}\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^{15} \times y^{15} = y^{15+15} = y^{30}\)[/tex]
- Result: [tex]\(140x^{12}y^{30}\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
#### Option 4:
- Dimensions: [tex]\(9x^4y^{11}\)[/tex] and [tex]\(12x^2y^4\)[/tex]
- Multiply the coefficients: [tex]\(9 \times 12 = 108\)[/tex]
- Multiply the [tex]\(x\)[/tex]-terms: [tex]\(x^4 \times x^2 = x^{4+2} = x^6\)[/tex]
- Multiply the [tex]\(y\)[/tex]-terms: [tex]\(y^{11} \times y^4 = y^{11+4} = y^{15}\)[/tex]
- Result: [tex]\(108x^6y^{15}\)[/tex] (not equal to [tex]\(24x^6y^{15}\)[/tex])
### Conclusion:
Of the given options, only Option 1 (with dimensions [tex]\(2x^5y^8\)[/tex] and [tex]\(12xy^7\)[/tex]) correctly multiplies to the given area [tex]\(24x^6y^{15}\)[/tex]. Thus, these are the possible dimensions of the rectangle.
