`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 expression \(3 \# 4^{5} + 3^{4} \# 5\), we will use the definitions provided in the question:

1. \(a \# b^{c} = (a \cdot b)^{c}\)
2. \(a^{b} \# c = a^{b} \cdot c\)

Now, let's break down the expression step by step.

### Step 1: Calculate \(3 \# 4^{5}\)

Using the first definition:
\[ 3 \# 4^{5} = (3 \cdot 4)^{5} \]

Calculating \(3 \cdot 4\):
\[ 3 \cdot 4 = 12 \]

Now raise it to the power of 5:
\[ 12^{5} \]

Calculating \(12^{5}\):
\[ 12^{5} = 248832 \]

### Step 2: Calculate \(3^{4} \# 5\)

Using the second definition:
\[ 3^{4} \# 5 = 3^{4} \cdot 5 \]

Calculating \(3^{4}\):
\[ 3^{4} = 81 \]

Now multiply by 5:
\[ 81 \cdot 5 = 405 \]

### Step 3: Combine the results

Now we add the two results together:
\[ 3 \# 4^{5} + 3^{4} \# 5 = 248832 + 405 \]

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

### Final Answer:
The value of \(3 \# 4^{5} + 3^{4} \# 5\) is \(249237\).

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