https://file.aiquickdraw.com/mathbot/file/1726145395621t70gmgem.jpg

The question in the image asks for the HCF (Highest Common Factor) of the expressions \(12xy^2\) and \(30x^2y\).

### Step-by-step solution:

1. **Prime Factorization of Constants:**
   - For 12: \(12 = 2^2 \times 3\)
   - For 30: \(30 = 2 \times 3 \times 5\)

2. **Variables:**
   - For \(12xy^2\), the variables are \(x\) and \(y^2\).
   - For \(30x^2y\), the variables are \(x^2\) and \(y\).

3. **Finding the HCF of constants:**
   - The common factors between 12 and 30 are \(2\) and \(3\), so the HCF of \(12\) and \(30\) is \(2 \times 3 = 6\).

4. **Finding the HCF of variables:**
   - For \(x\): The lowest power of \(x\) is \(x^1\).
   - For \(y\): The lowest power of \(y\) is \(y^1\).

### Final HCF:
Thus, the HCF of \(12xy^2\) and \(30x^2y\) is:
\[
6xy
\]

Let me know if you have any questions, or if you want further clarification.

Here are 5 related questions:
1. What is the LCM (Lowest Common Multiple) of \(12xy^2\) and \(30x^2y\)?
2. How do you find the HCF of three or more terms?
3. What happens if there are no common variables in the terms?
4. How do you factorize more complex algebraic expressions to find the HCF?
5. Can the HCF be larger than any of the terms involved?

### Tip: 
Always compare the lowest powers of common variables when determining the HCF of algebraic terms.

This detailed guide explains how to find the HCF of algebraic expressions like 12xy² and 30x²y using prime factorization and comparing the lowest powers of common variables. Suitable for Grades 8-10.

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