Find the value of `16^3`

Options

A. 4036
B. 4056
C. 4096
D. 4076

Answer

To find the value of \( 16^3 \), we can follow these steps:

### Step 1: Write the expression for \( 16^3 \)
We start with the expression:
\[ 16^3 \]

### Step 2: Break down \( 16 \)
We can express \( 16 \) as \( 20 - 4 \):
\[ 16 = 20 - 4 \]

### Step 3: Use the binomial expansion
Now we can apply the binomial expansion for \( (a - b)^3 \), where \( a = 20 \) and \( b = 4 \):
\[ (20 - 4)^3 = 20^3 - 3 \cdot 20^2 \cdot 4 + 3 \cdot 20 \cdot 4^2 - 4^3 \]

### Step 4: Calculate each term
1. Calculate \( 20^3 \):
\[ 20^3 = 8000 \]
2. Calculate \( 3 \cdot 20^2 \cdot 4 \):
\[ 20^2 = 400 \quad \Rightarrow \quad 3 \cdot 400 \cdot 4 = 4800 \]
3. Calculate \( 3 \cdot 20 \cdot 4^2 \):
\[ 4^2 = 16 \quad \Rightarrow \quad 3 \cdot 20 \cdot 16 = 960 \]
4. Calculate \( 4^3 \):
\[ 4^3 = 64 \]

### Step 5: Substitute back into the expansion
Now substituting these values back into the expansion:
\[ 16^3 = 8000 - 4800 + 960 - 64 \]

### Step 6: Perform the arithmetic
Now, we perform the arithmetic step-by-step:
1. First, subtract \( 4800 \) from \( 8000 \):
\[ 8000 - 4800 = 3200 \]
2. Next, add \( 960 \):
\[ 3200 + 960 = 4160 \]
3. Finally, subtract \( 64 \):
\[ 4160 - 64 = 4096 \]

### Final Result
Thus, the value of \( 16^3 \) is:
\[ \boxed{4096} \]