Express the following in the form a + ib
`3(7 + 7i) + i(7 + 7i)`

Answer

To express the given expression \(3(7 + 7i) + i(7 + 7i)\) in the form \(a + ib\), we will follow these steps:

### Step 1: Distribute the terms
First, we will distribute the 3 and \(i\) in the expression.

\[ 3(7 + 7i) = 3 \cdot 7 + 3 \cdot 7i = 21 + 21i \]

\[ i(7 + 7i) = i \cdot 7 + i \cdot 7i = 7i + 7i^2 \]

### Step 2: Substitute \(i^2\)
Recall that \(i^2 = -1\). Therefore, we can substitute \(i^2\) in the expression:

\[ 7i^2 = 7(-1) = -7 \]

So, we have:

\[ i(7 + 7i) = 7i - 7 \]

### Step 3: Combine the results
Now we can combine the results from Step 1 and Step 2:

\[ 3(7 + 7i) + i(7 + 7i) = (21 + 21i) + (7i - 7) \]

### Step 4: Combine like terms
Now we will combine the real parts and the imaginary parts:

Real parts: \(21 - 7 = 14\)

Imaginary parts: \(21i + 7i = 28i\)

So, we have:

\[ 14 + 28i \]

### Final Result
Thus, the expression \(3(7 + 7i) + i(7 + 7i)\) in the form \(a + ib\) is:

\[ 14 + 28i \]