Let `cancel(EE)` be an operation on a and b defined as a`cancel(EE)b=a^(b)`. If `p=4cancel(EE)`a, `q=4cancel(EE) b, r=2cancel(EE)` `(2c) and a+b+c=4`, find the value of pqr.

Options

A. 4
B. 16
C. 64
D. 256

Answer

To solve the problem step by step, we will define the operations and variables as given in the question, and then compute the value of \( pqr \).

### Step 1: Define the operation
The operation \( a \cancel{(EE)} b \) is defined as \( a^{b} \).

### Step 2: Define the variables
From the problem, we have:
- \( p = 4 \cancel{(EE)} a = 4^a \)
- \( q = 4 \cancel{(EE)} b = 4^b \)
- \( r = 2 \cancel{(EE)} (2c) = 2^{2c} \)

### Step 3: Calculate \( pqr \)
We need to find \( pqr \):
\[ pqr = (4^a) \cdot (4^b) \cdot (2^{2c}) \]
Using the property of exponents, we can combine \( 4^a \) and \( 4^b \):
\[ pqr = 4^{a+b} \cdot 2^{2c} \]

### Step 4: Express \( 4 \) in terms of \( 2 \)
Since \( 4 = 2^2 \), we can rewrite \( 4^{a+b} \):
\[ pqr = (2^2)^{a+b} \cdot 2^{2c} = 2^{2(a+b)} \cdot 2^{2c} = 2^{2(a+b + c)} \]

### Step 5: Use the given equation
We know from the problem statement that:
\[ a + b + c = 4 \]
Substituting this into our equation gives:
\[ pqr = 2^{2(4)} = 2^8 \]

### Step 6: Calculate \( 2^8 \)
Now, we calculate \( 2^8 \):
\[ 2^8 = 256 \]

### Final Answer
Thus, the value of \( pqr \) is:
\[ \boxed{256} \]