Find the specified term of the expression in each of the following binomials:
(i) Fifth term of `(2 a + 3b)^(12)`. Evaluate it when `a = (1)/(3), b = (1)/(4)`.

Answer

To find the fifth term of the expression \((2a + 3b)^{12}\) and evaluate it when \(a = \frac{1}{3}\) and \(b = \frac{1}{4}\), we can follow these steps:

### Step 1: Identify the formula for the term in a binomial expansion
The general term (T) in the expansion of \((x + y)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} x^{n-r} y^r \]
where:
- \(n\) is the exponent,
- \(r\) is the term number minus one,
- \(x\) and \(y\) are the terms in the binomial.

### Step 2: Assign values to the variables
In our case:
- \(x = 2a\),
- \(y = 3b\),
- \(n = 12\),
- We want the fifth term, so \(r = 4\) (since \(T_{5} = T_{r+1}\)).

### Step 3: Substitute the values into the formula
Using the formula, we can write the fifth term as:
\[ T_5 = \binom{12}{4} (2a)^{12-4} (3b)^4 \]

### Step 4: Calculate the binomial coefficient
Calculate \(\binom{12}{4}\):
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

### Step 5: Substitute and simplify the expression
Now substitute back into the term:
\[ T_5 = 495 (2a)^8 (3b)^4 \]
Calculating the powers:
\[ (2a)^8 = 2^8 a^8 = 256 a^8 \]
\[ (3b)^4 = 3^4 b^4 = 81 b^4 \]
Thus, we have:
\[ T_5 = 495 \cdot 256 a^8 \cdot 81 b^4 \]

### Step 6: Combine the coefficients
Now calculate the coefficient:
\[ 495 \cdot 256 \cdot 81 \]

### Step 7: Substitute \(a\) and \(b\) values
Now we substitute \(a = \frac{1}{3}\) and \(b = \frac{1}{4}\):
\[ T_5 = 495 \cdot 256 \cdot 81 \cdot \left(\frac{1}{3}\right)^8 \cdot \left(\frac{1}{4}\right)^4 \]

### Step 8: Calculate the powers of \(a\) and \(b\)
Calculating the powers:
\[ \left(\frac{1}{3}\right)^8 = \frac{1}{6561} \]
\[ \left(\frac{1}{4}\right)^4 = \frac{1}{256} \]

### Step 9: Combine everything
Now, substituting these values:
\[ T_5 = 495 \cdot 256 \cdot 81 \cdot \frac{1}{6561} \cdot \frac{1}{256} \]
The \(256\) cancels out:
\[ T_5 = 495 \cdot 81 \cdot \frac{1}{6561} \]

### Step 10: Final calculation
Calculating \(495 \cdot 81\):
\[ 495 \cdot 81 = 40095 \]
Thus,
\[ T_5 = \frac{40095}{6561} \]

### Step 11: Simplifying the fraction
Calculating \(40095 \div 6561\):
\[ T_5 = 5 \quad \text{(after simplification)} \]

### Final Answer
The fifth term of the expansion \((2a + 3b)^{12}\) evaluated at \(a = \frac{1}{3}\) and \(b = \frac{1}{4}\) is \(5\).