Question

Find the least common denominator.
StartFraction 5 Over 42 x cubed EndFraction
and StartFraction 7 Over 18 x EndFraction
Question content area bottom
Part 1
The least common denominator is
  
enter your response here.

Answer 1

To find the least common denominator (LCD) of the fractions \(\frac{5}{42x^3}\) and \(\frac{7}{18x}\), we need to find the least common multiple (LCM) of the denominators \(42x^3\) and \(18x\).

1. **Factor the denominators:**
- \(42 = 2 \times 3 \times 7\)
- \(18 = 2 \times 3^2\)

Incorporating \(x\):
- \(42x^3 = 2 \times 3 \times 7 \times x^3\)
- \(18x = 2 \times 3^2 \times x\)

2. **Find the LCM of the numerical coefficients:**
- The LCM takes the highest power of each prime factor:
- For \(2\), the max power between \(42\) and \(18\) is \(2^1\).
- For \(3\), the max power is \(3^2\).
- For \(7\), it is \(7^1\).

Thus, the LCM of \(42\) and \(18\) is:
\[
\text{LCM} = 2^1 \times 3^2 \times 7^1 = 2 \times 9 \times 7 = 126.
\]

3. **Combine the variable parts:**
- The highest power of \(x\) among \(x^3\) and \(x^1\) is \(x^3\).

4. **Combine the LCM of the coefficients with the variable part:**
- Therefore, the least common denominator (LCD) is:
\[
\text{LCD} = 126x^3.
\]

Thus, the least common denominator is \( \boxed{126x^3} \).