The HCF of `(x^(3)y)/(m^(2)n^(4)), (x^(2)y^(3))/(m^(2)n^(2))` and `(x^(4)y^(2))/(mn^(3))` is

Options

A. `(x^(2)y)/(mn^(2))`
B. `(x^(3)y^(2))/(mn^(2))`
C. `(x^(2)y)/(m^(2)n^(4))`
D. `(y x)/(m n^(2))`

Answer

To find the HCF (Highest Common Factor) of the three given expressions, we will follow these steps:

### Given Expressions:
1. \(\frac{x^3 y}{m^2 n^4}\)
2. \(\frac{x^2 y^3}{m^2 n^2}\)
3. \(\frac{x^4 y^2}{m n^3}\)

### Step 1: Identify the factors
The factors in the expressions are \(x\), \(y\), \(m\), and \(n\).

### Step 2: Find the least power of each factor
- **For \(x\)**:
- In the first expression: \(x^3\)
- In the second expression: \(x^2\)
- In the third expression: \(x^4\)

The least power of \(x\) is \(x^2\).

- **For \(y\)**:
- In the first expression: \(y^1\)
- In the second expression: \(y^3\)
- In the third expression: \(y^2\)

The least power of \(y\) is \(y^1\) (or simply \(y\)).

- **For \(m\)**:
- In the first expression: \(m^{-2}\) (since it is in the denominator)
- In the second expression: \(m^{-2}\)
- In the third expression: \(m^{-1}\)

The least power of \(m\) is \(m^{-2}\) (or \(\frac{1}{m^2}\)).

- **For \(n\)**:
- In the first expression: \(n^{-4}\)
- In the second expression: \(n^{-2}\)
- In the third expression: \(n^{-3}\)

The least power of \(n\) is \(n^{-4}\) (or \(\frac{1}{n^4}\)).

### Step 3: Combine the least powers
Now we combine the least powers of each factor to form the HCF:

\[ \text{HCF} = x^2 y m^{-2} n^{-4} = \frac{x^2 y}{m^2 n^4} \]

### Final Answer:
The HCF of the given expressions is \(\frac{x^2 y}{m^2 n^4}\).

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