If `a=2` and `b=3`, then find the value of
`[(1)/(a^b)+(1)/(b^a)]`.

Answer

To solve the expression \(\left[\frac{1}{a^b} + \frac{1}{b^a}\right]\) given \(a = 2\) and \(b = 3\), we will follow these steps:

### Step 1: Substitute the values of \(a\) and \(b\)
We start by substituting \(a = 2\) and \(b = 3\) into the expression:
\[ \frac{1}{a^b} + \frac{1}{b^a} = \frac{1}{2^3} + \frac{1}{3^2} \]

### Step 2: Calculate \(2^3\) and \(3^2\)
Next, we calculate the powers:
\[ 2^3 = 2 \times 2 \times 2 = 8 \]
\[ 3^2 = 3 \times 3 = 9 \]

### Step 3: Substitute the calculated values back into the expression
Now we substitute these values back into the expression:
\[ \frac{1}{2^3} + \frac{1}{3^2} = \frac{1}{8} + \frac{1}{9} \]

### Step 4: Find a common denominator
To add the fractions \(\frac{1}{8}\) and \(\frac{1}{9}\), we need a common denominator. The least common multiple of 8 and 9 is 72.

### Step 5: Convert each fraction to have the common denominator
Convert \(\frac{1}{8}\) and \(\frac{1}{9}\) to have a denominator of 72:
\[ \frac{1}{8} = \frac{9}{72} \quad (\text{since } 8 \times 9 = 72) \]
\[ \frac{1}{9} = \frac{8}{72} \quad (\text{since } 9 \times 8 = 72) \]

### Step 6: Add the fractions
Now we can add the two fractions:
\[ \frac{9}{72} + \frac{8}{72} = \frac{9 + 8}{72} = \frac{17}{72} \]

### Final Answer
Thus, the value of the expression \(\left[\frac{1}{a^b} + \frac{1}{b^a}\right]\) is:
\[ \frac{17}{72} \]

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