Question

the area of a rectangle is represented by the expression 6x^3y^5. which of the following could be the dimensions of the rectangle? 2x^3y and 3xy^4; 2x^3y and 3y^4; 2xy^2 and 4x^2y^3; 2xy^2 and 3x^2y^3

Explanation:

Step1: Recall area formula for rectangle

The area of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. We need to find two expressions whose product is $6x^{3}y^{5}$.

Step2: Multiply the first - pair of expressions

Multiply $2x^{3}y$ and $3xy^{4}$:
\[

$$\begin{align*} (2x^{3}y)\times(3xy^{4})&=(2\times3)\times(x^{3}\times x)\times(y\times y^{4})\\ & = 6\times x^{3 + 1}\times y^{1+4}\\ &=6x^{4}y^{5} \end{align*}$$

\]

Step3: Multiply the second - pair of expressions

Multiply $2x^{3}y$ and $3y^{4}$:
\[

$$\begin{align*} (2x^{3}y)\times(3y^{4})&=(2\times3)\times x^{3}\times(y\times y^{4})\\ &=6x^{3}y^{5} \end{align*}$$

\]

Step4: Multiply the third - pair of expressions

Multiply $2xy^{2}$ and $4x^{2}y^{3}$:
\[

$$\begin{align*} (2xy^{2})\times(4x^{2}y^{3})&=(2\times4)\times(x\times x^{2})\times(y^{2}\times y^{3})\\ &=8x^{3}y^{5} \end{align*}$$

\]

Step5: Multiply the fourth - pair of expressions

Multiply $2xy^{2}$ and $3x^{2}y^{3}$:
\[

$$\begin{align*} (2xy^{2})\times(3x^{2}y^{3})&=(2\times3)\times(x\times x^{2})\times(y^{2}\times y^{3})\\ &=6x^{3}y^{5} \end{align*}$$

\]

Answer:

$2x^{3}y$ and $3y^{4}$; $2xy^{2}$ and $3x^{2}y^{3}$