The H.C.F. of `14x^(3)y^(2) " and " 21y^(3)z^(2)` is

Options

A. 7xyz
B. `7x^(2)y`
C. `7y^(2)`
D. xyz

Answer

To find the H.C.F. (Highest Common Factor) of the algebraic expressions \(14x^3y^2\) and \(21y^3z^2\), we will follow these steps:

### Step 1: Factor the coefficients
First, we need to factor the numerical coefficients of both expressions.

- For \(14\):
\[ 14 = 2 \times 7 \]

- For \(21\):
\[ 21 = 3 \times 7 \]

### Step 2: Identify the common factors of the coefficients
Now, we look for the common factors in the coefficients \(14\) and \(21\).

- The common factor is \(7\).

### Step 3: Factor the variables
Next, we will factor the variable parts of both expressions.

- The first expression \(14x^3y^2\) has:
- \(x^3\) (which means \(x\) appears 3 times)
- \(y^2\) (which means \(y\) appears 2 times)

- The second expression \(21y^3z^2\) has:
- \(y^3\) (which means \(y\) appears 3 times)
- \(z^2\) (which means \(z\) appears 2 times)

### Step 4: Identify the common variables
Now, we identify the common variables and their lowest powers.

- The common variable is \(y\).
- The lowest power of \(y\) in both expressions is \(y^2\).

### Step 5: Combine the common factors
Now, we combine the common numerical factor and the common variable factor to find the H.C.F.

\[ \text{H.C.F.} = 7y^2 \]

### Final Answer
Thus, the H.C.F. of \(14x^3y^2\) and \(21y^3z^2\) is:
\[ \boxed{7y^2} \]