We are tasked with finding the length and width of a rectangle, where the area is given by:

$70y^8 + 30y^6$

The width is equal to the greatest common monomial factor of $70y^8$ and $30y^6$.

Step 1: Find the greatest common monomial factor (GCF)

The GCF of the coefficients $70$ and $30$ is $10$ because:

$\text{GCF of } 70 \text{ and } 30 = 10$

For the variable terms, we take the lowest power of $y$ that is common in both terms. The lowest power of $y$ is $y^6$, since:

$\text{GCF of } y^8 \text{ and } y^6 = y^6$

Thus, the greatest common monomial factor is:

$10y^6$

This is the width of the rectangle.

Step 2: Factor the expression for the length

Now, we can factor out $10y^6$ from the expression for the area:

$70y^8 + 30y^6 = 10y^6(7y^2 + 3)$

Thus, the length of the rectangle is:

$7y^2 + 3$

Final Answer:

• Width = $10y^6$
• Length = $7y^2 + 3$

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do you find the greatest common factor of polynomials?
2. What is the difference between factoring and expanding polynomials?
3. How does the distributive property apply when factoring?
4. Can the area of a rectangle always be expressed as a product of its length and width in algebra?
5. How do exponents influence the greatest common factor?

Tip: When factoring polynomials, always check both the coefficients and the variables for common factors.