Evaluate `(a+2b)^(3)`.

Options

A. `a^(3)+6a^(2)b+12ab^(2)+8b^(3)`
B. `a^(3)-6a^(2)b-12ab^(2)+8b^(3)`
C. `a^(3)-6a^(2)b+12ab^(2)-8b^(3)`
D. None of These

Answer

To evaluate \((a + 2b)^3\), we can use the identity for the cube of a binomial, which states:

\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

In our case, we have \(a\) and \(b\) where \(b\) is replaced by \(2b\). Therefore, we can rewrite the expression as:

\[ (a + 2b)^3 = a^3 + (2b)^3 + 3a^2(2b) + 3a(2b)^2 \]

Now, let's break this down step by step:

### Step 1: Calculate \(a^3\)
This remains as \(a^3\).

### Step 2: Calculate \((2b)^3\)
\[ (2b)^3 = 2^3 \cdot b^3 = 8b^3 \]

### Step 3: Calculate \(3a^2(2b)\)
\[ 3a^2(2b) = 6a^2b \]

### Step 4: Calculate \(3a(2b)^2\)
\[ (2b)^2 = 4b^2 \quad \text{so,} \quad 3a(2b)^2 = 3a \cdot 4b^2 = 12ab^2 \]

### Step 5: Combine all terms
Now, we can combine all the calculated terms together:
\[ (a + 2b)^3 = a^3 + 8b^3 + 6a^2b + 12ab^2 \]

Thus, the final result is:
\[ \boxed{a^3 + 8b^3 + 6a^2b + 12ab^2} \]