Find the least common denominator (LCD) of the fractions $\frac{4}{45xy^5}$ and $\frac{8}{45x^5y^3}$.

1. Problem Description

Find the least common denominator (LCD) of the fractions $\frac{4}{45xy^5}$ and $\frac{8}{45x^5y^3}$.

2. Solution Steps

To find the LCD of the given fractions, we need to find the least common multiple (LCM) of the denominators $45xy^5$ and $45x^5y^3$.

First, we find the LCM of the coefficients. The coefficient in both denominators is

4

5. Therefore the LCM of 45 and 45 is

4

5.

Next, we find the LCM of the variable $x$. We have $x$ and $x^5$.

The LCM is $x^5$, as it is the highest power of $x$ in either denominator.

Then, we find the LCM of the variable $y$. We have $y^5$ and $y^3$.

The LCM is $y^5$, as it is the highest power of $y$ in either denominator.

Therefore, the LCD is $45x^5y^5$.

3. Final Answer

$45x^5y^5$