Answer :
We are given that the area of the rectangle is
[tex]$$
A = 24x^6y^{15}.
$$[/tex]
If the length and width are represented by two expressions, their product must equal the area. Let’s check the first option.
The first option gives the dimensions as
[tex]$$
\text{Dimension 1} = 2x^5y^8 \quad \text{and} \quad \text{Dimension 2} = 12xy^7.
$$[/tex]
Multiplying these together, we have
[tex]$$
(2x^5y^8)(12xy^7) = (2 \cdot 12)(x^5 \cdot x)(y^8 \cdot y^7).
$$[/tex]
Let’s multiply the coefficients first:
[tex]$$
2 \cdot 12 = 24.
$$[/tex]
Next, add the exponents for [tex]$x$[/tex]:
[tex]$$
x^5 \cdot x = x^{5+1} = x^6.
$$[/tex]
And add the exponents for [tex]$y$[/tex]:
[tex]$$
y^8 \cdot y^7 = y^{8+7} = y^{15}.
$$[/tex]
Thus, the product is
[tex]$$
24x^6y^{15},
$$[/tex]
which exactly matches the given area.
Since the product of the dimensions in the first option reproduces the area, the dimensions of the rectangle can be
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
[tex]$$
A = 24x^6y^{15}.
$$[/tex]
If the length and width are represented by two expressions, their product must equal the area. Let’s check the first option.
The first option gives the dimensions as
[tex]$$
\text{Dimension 1} = 2x^5y^8 \quad \text{and} \quad \text{Dimension 2} = 12xy^7.
$$[/tex]
Multiplying these together, we have
[tex]$$
(2x^5y^8)(12xy^7) = (2 \cdot 12)(x^5 \cdot x)(y^8 \cdot y^7).
$$[/tex]
Let’s multiply the coefficients first:
[tex]$$
2 \cdot 12 = 24.
$$[/tex]
Next, add the exponents for [tex]$x$[/tex]:
[tex]$$
x^5 \cdot x = x^{5+1} = x^6.
$$[/tex]
And add the exponents for [tex]$y$[/tex]:
[tex]$$
y^8 \cdot y^7 = y^{8+7} = y^{15}.
$$[/tex]
Thus, the product is
[tex]$$
24x^6y^{15},
$$[/tex]
which exactly matches the given area.
Since the product of the dimensions in the first option reproduces the area, the dimensions of the rectangle can be
[tex]$$
2x^5y^8 \quad \text{and} \quad 12xy^7.
$$[/tex]
