Let `A= R- {-1}`. Let `**` be defined on A as `a ** b= a + b+ ab`, for all `a, b in A`. Solve the equation `2 **x **5=4`

Options

A. `(13)/(18)`
B. `-(18)/(13)`
C. `-(13)/(18)`
D. None of these

Answer

To solve the equation \(2 ** x ** 5 = 4\) where the operation \(**\) is defined as \(a ** b = a + b + ab\), we will follow these steps:

### Step 1: Understand the operation
The operation \(a ** b\) can be rewritten as:
\[ a ** b = a + b + ab \]
This means that for any two elements \(a\) and \(b\) in the set \(A\), the operation combines them in a specific way.

### Step 2: Rewrite the equation
We start with the equation:
\[ 2 ** x ** 5 = 4 \]
We can first calculate \(2 ** x\) and then use that result to calculate \((2 ** x) ** 5\).

### Step 3: Calculate \(2 ** x\)
Using the definition of the operation:
\[ 2 ** x = 2 + x + 2x = 2 + x + 2x = 2 + 3x \]

### Step 4: Calculate \((2 ** x) ** 5\)
Now we need to calculate \((2 ** x) ** 5\):
\[ (2 ** x) ** 5 = (2 + 3x) ** 5 \]
Using the operation definition again:
\[ (2 + 3x) ** 5 = (2 + 3x) + 5 + (2 + 3x) \cdot 5 \]
Calculating the multiplication:
\[ (2 + 3x) \cdot 5 = 10 + 15x \]
Thus,
\[ (2 + 3x) ** 5 = (2 + 3x) + 5 + (10 + 15x) = 2 + 3x + 5 + 10 + 15x \]
Combining like terms:
\[ = 17 + 18x \]

### Step 5: Set the equation to 4
Now we have:
\[ 17 + 18x = 4 \]

### Step 6: Solve for \(x\)
Subtract 17 from both sides:
\[ 18x = 4 - 17 \]
\[ 18x = -13 \]
Now divide by 18:
\[ x = -\frac{13}{18} \]

### Final Answer
Thus, the solution to the equation \(2 ** x ** 5 = 4\) is:
\[ x = -\frac{13}{18} \]