Consider the following pattern:
`2^(3)-1^(3)=1+2 xx 1 xx 3`
`3^(3)-2^(3)=1+3 xx 2 xx 3`
`4^(3)-3^(3)=1+4 xx 3 xx 3`
Using the above pattern,find the value of the following :
`7^(3)-6^(3)` .

Answer

To solve the problem \(7^3 - 6^3\) using the given pattern, we can follow these steps:

### Step 1: Identify the pattern
From the examples provided:
- \(2^3 - 1^3 = 1 + 2 \times 1 \times 3\)
- \(3^3 - 2^3 = 1 + 3 \times 2 \times 3\)
- \(4^3 - 3^3 = 1 + 4 \times 3 \times 3\)

We observe that for \(n^3 - (n-1)^3\), the formula can be expressed as:
\[ n^3 - (n-1)^3 = 1 + n \times (n-1) \times n \]

### Step 2: Apply the pattern to \(7^3 - 6^3\)
Using \(n = 7\):
\[ 7^3 - 6^3 = 1 + 7 \times 6 \times 7 \]

### Step 3: Calculate the multiplication
Now we need to calculate \(7 \times 6 \times 7\):
\[ 7 \times 6 = 42 \]
\[ 42 \times 7 = 294 \]

### Step 4: Add 1 to the result
Now, we add 1 to the result we obtained:
\[ 1 + 294 = 295 \]

### Final Answer
Thus, the value of \(7^3 - 6^3\) is:
\[ \boxed{295} \]

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