Find the value using the short-cut method:
`(84)^(3)`

Answer

To find the value of \( (84)^3 \) using the shortcut method, we will use the formula for the cube of a binomial, which is:

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

### Step-by-Step Solution:

1. **Identify \( a \) and \( b \)**:
- Let \( a = 80 \) and \( b = 4 \) (since \( 84 = 80 + 4 \)).

2. **Calculate \( a^3 \)**:
- \( a^3 = (80)^3 = 512000 \) (since \( 80^3 = 512 \times 1000 \)).

3. **Calculate \( b^3 \)**:
- \( b^3 = (4)^3 = 64 \).

4. **Calculate \( 3a^2b \)**:
- First, find \( a^2 = (80)^2 = 6400 \).
- Then, calculate \( 3a^2b = 3 \times 6400 \times 4 = 76800 \).

5. **Calculate \( 3ab^2 \)**:
- First, find \( b^2 = (4)^2 = 16 \).
- Then, calculate \( 3ab^2 = 3 \times 80 \times 16 = 3840 \).

6. **Combine all the terms**:
- Now, we will sum all the calculated values:
\[ (80 + 4)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]
\[ = 512000 + 64 + 76800 + 3840 \]

7. **Perform the addition**:
- First, add \( 512000 + 64 = 512064 \).
- Next, add \( 512064 + 76800 = 588864 \).
- Finally, add \( 588864 + 3840 = 592704 \).

### Final Answer:
Thus, the value of \( (84)^3 \) is \( 592704 \).

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