a*b#c = `(ab)^c` :
a#b*c = `a^b`.c, then find the value of 3#4*5+3*4#5

Options

A. 35287
B. 249237
C. 12345
D. none of these

Answer

To solve the problem, we need to evaluate the expression \(3 \# 4 * 5 + 3 * 4 \# 5\) using the definitions provided for the operations \(a * b \# c\) and \(a \# b * c\).

### Step 1: Evaluate \(3 \# 4 * 5\)

According to the definition:
\[ a \# b * c = a^b \cdot c \]
Substituting \(a = 3\), \(b = 4\), and \(c = 5\):
\[ 3 \# 4 * 5 = 3^4 \cdot 5 \]

Now, calculate \(3^4\):
\[ 3^4 = 81 \]
Thus,
\[ 3 \# 4 * 5 = 81 \cdot 5 = 405 \]

### Step 2: Evaluate \(3 * 4 \# 5\)

According to the definition:
\[ a * b \# c = (ab)^c \]
Substituting \(a = 3\), \(b = 4\), and \(c = 5\):
\[ 3 * 4 \# 5 = (3 \cdot 4)^5 \]

Now, calculate \(3 \cdot 4\):
\[ 3 \cdot 4 = 12 \]
Thus,
\[ 3 * 4 \# 5 = 12^5 \]

Now, calculate \(12^5\):
\[ 12^5 = 248832 \]

### Step 3: Combine the results

Now we need to add the results from Step 1 and Step 2:
\[ 3 \# 4 * 5 + 3 * 4 \# 5 = 405 + 248832 \]

Calculating the sum:
\[ 405 + 248832 = 249237 \]

### Final Answer
Thus, the value of \(3 \# 4 * 5 + 3 * 4 \# 5\) is:
\[ \boxed{249237} \]