Find the values of the following by using suitable identity:
`(206)^(3)`

Answer

To find the value of \( (206)^3 \) using a suitable identity, we can use the algebraic identity for the cube of a binomial:

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

### Step 1: Identify \( a \) and \( b \)
We can express \( 206 \) as \( 200 + 6 \). Here, we can set:
- \( a = 200 \)
- \( b = 6 \)

### Step 2: Apply the identity
Using the identity, we have:
\[ (200 + 6)^3 = 200^3 + 6^3 + 3 \cdot 200 \cdot 6 \cdot (200 + 6) \]

### Step 3: Calculate \( a^3 \) and \( b^3 \)
Now we calculate \( a^3 \) and \( b^3 \):
- \( 200^3 = 8,000,000 \) (since \( 200 = 2 \times 10^2 \) and \( (2 \times 10^2)^3 = 2^3 \times 10^6 = 8 \times 10^6 \))
- \( 6^3 = 216 \)

### Step 4: Calculate \( 3ab(a + b) \)
Next, we calculate \( 3ab(a + b) \):
- \( a + b = 200 + 6 = 206 \)
- \( 3ab = 3 \cdot 200 \cdot 6 = 3600 \)

Now, we multiply:
\[ 3ab(a + b) = 3600 \cdot 206 \]

Calculating \( 3600 \cdot 206 \):
\[ 3600 \cdot 206 = 739200 \]

### Step 5: Combine all parts
Now we combine all parts:
\[ (206)^3 = 200^3 + 6^3 + 3ab(a + b) = 8,000,000 + 216 + 739200 \]

### Step 6: Perform the final addition
Now we perform the addition:
\[ 8,000,000 + 739200 + 216 = 8,741,216 \]

Thus, the value of \( (206)^3 \) is:
\[ \boxed{87481216} \]