If `4^a + 4^(a+1) = 4^(a + 2) - 176`, what is the value of a?

Answer

To solve the equation \( 4^a + 4^{a+1} = 4^{a+2} - 176 \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ 4^a + 4^{a+1} = 4^{a+2} - 176 \]
We can express \( 4^{a+1} \) as \( 4^a \cdot 4 \) and \( 4^{a+2} \) as \( 4^a \cdot 4^2 \):
\[ 4^a + 4 \cdot 4^a = 16 \cdot 4^a - 176 \]

### Step 2: Factor out \( 4^a \)
Now, factor out \( 4^a \) from the left side:
\[ 4^a (1 + 4) = 16 \cdot 4^a - 176 \]
This simplifies to:
\[ 5 \cdot 4^a = 16 \cdot 4^a - 176 \]

### Step 3: Rearrange the equation
Next, rearrange the equation to isolate terms involving \( 4^a \):
\[ 5 \cdot 4^a - 16 \cdot 4^a = -176 \]
This simplifies to:
\[ -11 \cdot 4^a = -176 \]

### Step 4: Divide both sides by -11
Now, divide both sides by -11:
\[ 4^a = \frac{176}{11} \]
Calculating the right side gives:
\[ 4^a = 16 \]

### Step 5: Rewrite \( 16 \) as a power of \( 4 \)
Since \( 16 = 4^2 \), we can equate the exponents:
\[ 4^a = 4^2 \]
Thus, we have:
\[ a = 2 \]

### Final Answer
The value of \( a \) is:
\[ \boxed{2} \]