Find the value using the short-cut method:
`(25) ^3`

Options

A. `15625`
B. `15635`
C. `15645`
D. `15655`

Answer

To find the value of \( (25)^3 \) using the shortcut method, we can apply the formula for the cube of a binomial, which is \( (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).

### Step-by-Step Solution:

1. **Express 25 as a sum**:
\[ 25 = 20 + 5 \]

2. **Identify \( a \) and \( b \)**:
Here, \( a = 20 \) and \( b = 5 \).

3. **Apply the binomial cube formula**:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]

4. **Calculate \( a^3 \)**:
\[ a^3 = 20^3 = 8000 \]

5. **Calculate \( b^3 \)**:
\[ b^3 = 5^3 = 125 \]

6. **Calculate \( 3a^2b \)**:
\[ 3a^2b = 3 \times (20^2) \times 5 = 3 \times 400 \times 5 = 6000 \]

7. **Calculate \( 3ab^2 \)**:
\[ 3ab^2 = 3 \times 20 \times (5^2) = 3 \times 20 \times 25 = 1500 \]

8. **Combine all parts**:
\[ (25)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = 8000 + 6000 + 1500 + 125 \]

9. **Perform the addition**:
\[ 8000 + 6000 = 14000 \]
\[ 14000 + 1500 = 15500 \]
\[ 15500 + 125 = 15625 \]

### Final Answer:
\[ (25)^3 = 15625 \]