If `a =2, b = 3,` then `(a ^(b) + b ^(a)) ^(-1)` is :

Options

A. `(1)/(31)`
B. `(1)/(17)`
C. `(1)/(21)`
D. `(1)/(13)`

Answer

To solve the expression \((a^{b} + b^{a})^{-1}\) given \(a = 2\) and \(b = 3\), we can follow these steps:

### Step 1: Substitute the values of \(a\) and \(b\)
We start by substituting \(a = 2\) and \(b = 3\) into the expression.

\[ (2^{3} + 3^{2})^{-1} \]

### Step 2: Calculate \(2^{3}\)
Now, we calculate \(2^{3}\):

\[ 2^{3} = 2 \times 2 \times 2 = 8 \]

### Step 3: Calculate \(3^{2}\)
Next, we calculate \(3^{2}\):

\[ 3^{2} = 3 \times 3 = 9 \]

### Step 4: Add the results
Now, we add the results of \(2^{3}\) and \(3^{2}\):

\[ 8 + 9 = 17 \]

### Step 5: Take the reciprocal
Finally, we take the reciprocal of the sum:

\[ (2^{3} + 3^{2})^{-1} = \frac{1}{17} \]

Thus, the final answer is:

\[ \frac{1}{17} \]