Answer :
We are given that the area of a rectangle is expressed as
[tex]$$
24x^6y^{15},
$$[/tex]
and the area is obtained by multiplying the dimensions (length and width) of the rectangle. One candidate pair of dimensions is
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
We'll show step by step that when these two expressions are multiplied, they produce the target area.
1. Multiply the coefficients:
[tex]$$
2 \times 12 = 24.
$$[/tex]
2. Multiply the [tex]$x$[/tex] terms. Remember, when multiplying powers of the same base, we add the exponents:
[tex]$$
x^5 \times x^1 = x^{5+1}= x^6.
$$[/tex]
3. Multiply the [tex]$y$[/tex] terms in a similar way:
[tex]$$
y^8 \times y^7 = y^{8+7}= y^{15}.
$$[/tex]
4. Combine the results from the above multiplications:
[tex]$$
24 \times x^6 \times y^{15} = 24x^6y^{15}.
$$[/tex]
Since this product exactly matches the given area expression, the correct dimensions of the rectangle are
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
Thus, the correct answer is the first option.
[tex]$$
24x^6y^{15},
$$[/tex]
and the area is obtained by multiplying the dimensions (length and width) of the rectangle. One candidate pair of dimensions is
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
We'll show step by step that when these two expressions are multiplied, they produce the target area.
1. Multiply the coefficients:
[tex]$$
2 \times 12 = 24.
$$[/tex]
2. Multiply the [tex]$x$[/tex] terms. Remember, when multiplying powers of the same base, we add the exponents:
[tex]$$
x^5 \times x^1 = x^{5+1}= x^6.
$$[/tex]
3. Multiply the [tex]$y$[/tex] terms in a similar way:
[tex]$$
y^8 \times y^7 = y^{8+7}= y^{15}.
$$[/tex]
4. Combine the results from the above multiplications:
[tex]$$
24 \times x^6 \times y^{15} = 24x^6y^{15}.
$$[/tex]
Since this product exactly matches the given area expression, the correct dimensions of the rectangle are
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
Thus, the correct answer is the first option.
