To find the least common denominator (LCD) of the given fractions:
[tex]\[
\frac{1}{45 x^3 y^6} \quad \text{and} \quad \frac{7}{30 x^5 y^4},
\][/tex]
we need to determine the least common multiple (LCM) of their denominators.
### Step-by-Step Solution
1. Identify the coefficients of the denominators:
- For the first fraction, the coefficient is 45.
- For the second fraction, the coefficient is 30.
2. Find the LCM of the coefficients 45 and 30:
- The factors of 45 are \(3^2 \times 5\).
- The factors of 30 are \(2 \times 3 \times 5\).
- The LCM is found by taking the highest power of each prime factor that appears in the factorizations:
- Factor of 2: \(2^1\)
- Factor of 3: \(3^2\)
- Factor of 5: \(5^1\)
- Thus, the LCM of 45 and 30 is \(2^1 \times 3^2 \times 5^1 = 90\).
3. Identify the exponents of the variables \(x\) and \(y\) in each denominator:
- For \(x\), in the first fraction it is \(x^3\), and in the second fraction it is \(x^5\).
- For \(y\), in the first fraction it is \(y^6\), and in the second fraction it is \(y^4\).
4. Determine the LCM of the variable parts:
- For \(x\), take the highest power appearing in the denominators, which is \(x^5\).
- For \(y\), take the highest power appearing in the denominators, which is \(y^6\).
5. Combine the results to form the LCD:
- The least common denominator is the product of the LCM of the coefficients and the highest powers of \(x\) and \(y\).
Therefore, the LCD is:
[tex]\[
90 \cdot x^5 \cdot y^6
\][/tex]
Thus, the least common denominator (LCD) for the fractions \(\frac{1}{45 x^3 y^6}\) and \(\frac{7}{30 x^5 y^4}\) is:
[tex]\[
\boxed{90 \cdot x^5 \cdot y^6}
\][/tex]
