The problem asks us to find the least common denominator (LCD) of the two fractions $\frac{7}{45xy^3}$ and $\frac{8}{45x^5y^2}$.

1. Problem Description

The problem asks us to find the least common denominator (LCD) of the two fractions $\frac{7}{45xy^3}$ and $\frac{8}{45x^5y^2}$.

2. Solution Steps

To find the LCD of two fractions, we need to find the least common multiple (LCM) of their denominators. The denominators are $45xy^3$ and $45x^5y^2$.

First, find the prime factorization of the coefficients: $45 = 3^2 \cdot 5$.

The denominators are $3^2 \cdot 5 \cdot x^1 \cdot y^3$ and $3^2 \cdot 5 \cdot x^5 \cdot y^2$.

To find the LCM, we take the highest power of each prime factor present in the denominators.

The LCM of the numerical coefficients is

4

5. For $x$, we have $x^1$ and $x^5$, so we take $x^5$.

For $y$, we have $y^3$ and $y^2$, so we take $y^3$.

Therefore, the LCD is $45x^5y^3$.

3. Final Answer

$45x^5y^3$