Answer :
We are given that the area of a rectangle is expressed as
[tex]$$24x^6y^{15},$$[/tex]
and we need to determine which set of dimensions (length and width) would give this area when multiplied.
Let the dimensions be [tex]$L$[/tex] and [tex]$W$[/tex]. Their product must equal the area:
[tex]$$L \times W = 24x^6y^{15}.$$[/tex]
Now, consider the dimensions:
[tex]$$L = 2x^5y^8 \quad \text{and} \quad W = 12xy^7.$$[/tex]
We multiply these to check if they produce the given area:
[tex]\[
\begin{aligned}
(2x^5y^8) \times (12xy^7) &= (2 \times 12) \times (x^5 \times x) \times (y^8 \times y^7) \\
&= 24 \times x^{5+1} \times y^{8+7} \\
&= 24x^6y^{15}.
\end{aligned}
\][/tex]
Since the product equals the given area, the dimensions are correct.
Thus, the dimensions of the rectangle are
[tex]$$\boxed{2x^5y^8 \text{ and } 12xy^7.}$$[/tex]
[tex]$$24x^6y^{15},$$[/tex]
and we need to determine which set of dimensions (length and width) would give this area when multiplied.
Let the dimensions be [tex]$L$[/tex] and [tex]$W$[/tex]. Their product must equal the area:
[tex]$$L \times W = 24x^6y^{15}.$$[/tex]
Now, consider the dimensions:
[tex]$$L = 2x^5y^8 \quad \text{and} \quad W = 12xy^7.$$[/tex]
We multiply these to check if they produce the given area:
[tex]\[
\begin{aligned}
(2x^5y^8) \times (12xy^7) &= (2 \times 12) \times (x^5 \times x) \times (y^8 \times y^7) \\
&= 24 \times x^{5+1} \times y^{8+7} \\
&= 24x^6y^{15}.
\end{aligned}
\][/tex]
Since the product equals the given area, the dimensions are correct.
Thus, the dimensions of the rectangle are
[tex]$$\boxed{2x^5y^8 \text{ and } 12xy^7.}$$[/tex]
