Find the value using the short-cut method:
`(68)^(3)`

Answer

To find the value of \( (68)^3 \) using the shortcut method, we can express 68 in a way that simplifies our calculations. Here’s a step-by-step solution:

### Step 1: Express 68 in a convenient form
We can express 68 as:
\[ 68 = 60 + 8 \]

### Step 2: Use the cube expansion formula
We will use the formula for the cube of a binomial:
\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]
Here, \( a = 60 \) and \( b = 8 \).

### Step 3: Calculate \( a^3 \) and \( b^3 \)
Calculate \( 60^3 \):
\[ 60^3 = (6 \times 10)^3 = 6^3 \times 10^3 = 216 \times 1000 = 216000 \]

Calculate \( 8^3 \):
\[ 8^3 = 512 \]

### Step 4: Calculate \( 3a^2b \)
Calculate \( 3 \times 60^2 \times 8 \):
\[ 60^2 = 3600 \]
\[ 3 \times 3600 \times 8 = 3 \times 28800 = 86400 \]

### Step 5: Calculate \( 3ab^2 \)
Calculate \( 3 \times 60 \times 8^2 \):
\[ 8^2 = 64 \]
\[ 3 \times 60 \times 64 = 3 \times 3840 = 11520 \]

### Step 6: Combine all the parts
Now, we combine all the calculated values:
\[ (68)^3 = 60^3 + 8^3 + 3 \times 60^2 \times 8 + 3 \times 60 \times 8^2 \]
\[ = 216000 + 512 + 86400 + 11520 \]

### Step 7: Perform the final addition
Now, let's add these values together:
\[ 216000 + 512 = 216512 \]
\[ 216512 + 86400 = 302912 \]
\[ 302912 + 11520 = 314432 \]

Thus, the value of \( (68)^3 \) is:
\[ \boxed{314432} \]