Given : 1176 = `2^(p) . 3^(q) . 7^(r )`. Find :
the numerical values of p,q and r.

Answer

To solve the equation \( 1176 = 2^p \cdot 3^q \cdot 7^r \) and find the values of \( p \), \( q \), and \( r \), we will perform the prime factorization of 1176 step-by-step.

### Step 1: Prime Factorization of 1176
We start by dividing 1176 by the smallest prime number, which is 2.

1. \( 1176 \div 2 = 588 \)
2. \( 588 \div 2 = 294 \)
3. \( 294 \div 2 = 147 \)

Now, 147 is not divisible by 2, so we move to the next prime number, which is 3.

4. \( 147 \div 3 = 49 \)

Next, we divide 49 by the next prime number, which is 7.

5. \( 49 \div 7 = 7 \)
6. \( 7 \div 7 = 1 \)

Now we have completely factored 1176 into prime factors.

### Step 2: Write the Prime Factorization
From the steps above, we can write the prime factorization of 1176 as:
\[ 1176 = 2^3 \cdot 3^1 \cdot 7^2 \]

### Step 3: Equate with the Given Expression
Now we can equate this with the expression \( 2^p \cdot 3^q \cdot 7^r \):
\[ 2^p \cdot 3^q \cdot 7^r = 2^3 \cdot 3^1 \cdot 7^2 \]

### Step 4: Compare the Exponents
Now we compare the exponents of the same bases:
- For base 2: \( p = 3 \)
- For base 3: \( q = 1 \)
- For base 7: \( r = 2 \)

### Final Values
Thus, the values are:
- \( p = 3 \)
- \( q = 1 \)
- \( r = 2 \)

### Summary of the Solution
The numerical values of \( p \), \( q \), and \( r \) are:
- \( p = 3 \)
- \( q = 1 \)
- \( r = 2 \)