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To find the least common denominator (LCD) for the fractions given, we need to determine the least common multiple (LCM) of the denominators.

The denominators are:
1. \( 4xy^3z \)
2. \( 8x^2y \)
3. \( 12yz^2 \)

### Step-by-step process:

1. **Numerical Coefficients**:
   - The numerical coefficients are \( 4 \), \( 8 \), and \( 12 \).
   - The least common multiple of \( 4 \), \( 8 \), and \( 12 \) is \( 24 \).

2. **For \( x \)**:
   - The powers of \( x \) in the denominators are \( x^1 \), \( x^2 \), and \( x^0 \) (since \( x \) does not appear in the third denominator).
   - To cover all the \( x \)'s, we take the highest power, which is \( x^2 \).

3. **For \( y \)**:
   - The powers of \( y \) are \( y^3 \), \( y^1 \), and \( y^1 \).
   - The highest power of \( y \) is \( y^3 \).

4. **For \( z \)**:
   - The powers of \( z \) are \( z^1 \), \( z^0 \) (since \( z \) does not appear in the second denominator), and \( z^2 \).
   - The highest power of \( z \) is \( z^2 \).

### Least Common Denominator:
Thus, the least common denominator (LCD) is:

\[
24x^2y^3z^2
\]

Let me know if you need further details!

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### Related Questions:
1. How is the least common multiple (LCM) different from the greatest common divisor (GCD)?
2. How do you find the LCM of variables in algebraic fractions?
3. Can the least common denominator include negative values?
4. What happens when the exponents of variables are equal across all terms in a denominator?
5. How do you simplify fractions once the least common denominator is found?

### Tip:
Always break down each part of the denominator (numerical coefficient and variables) and work with the highest powers when finding the least common denominator.

Learn how to determine the least common denominator (LCD) of algebraic fractions like 3/(4xy^3z), -7/(8x^2y), and 5/(12yz^2). Follow a step-by-step guide involving the least common multiple (LCM) and the highest powers of variables. Suitable for students in Grades 7-10.

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