Define be operation symbols a*b=(a+b)*(a-1)*(b-1), find the value of 4*(5*6)


To solve the problem, we need to use the given operation symbol defined as \( a * b = (a+b) \times (a-1) \times (b-1) \).

### Step 1: Calculate \( 5 * 6 \)

Using the operation definition \( a * b = (a+b) \times (a-1) \times (b-1) \):

\[
5 * 6 = (5 + 6) \times (5 - 1) \times (6 - 1)
\]

\[
5 * 6 = 11 \times 4 \times 5
\]

\[
5 * 6 = 220
\]

### Step 2: Calculate \( 4 * (5 * 6) \)

Now that we know \( 5 * 6 = 220 \), we can calculate \( 4 * 220 \) using the same operation definition:

\[
4 * 220 = (4 + 220) \times (4 - 1) \times (220 - 1)
\]

\[
4 * 220 = 224 \times 3 \times 219
\]

\[
4 * 220 = 672 \times 219
\]

\[
4 * 220 = 147,168
\]

Thus, the value of \( 4 * (5 * 6) \) is **147,168**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the value of \( 3 * (2 * 4) \) using the same operation definition?
2. How would you define an inverse operation for \( a * b = (a+b) \times (a-1) \times (b-1) \)?
3. Can you determine a pattern for the results of \( a * b \) for small integer values of \( a \) and \( b \)?
4. If the operation were defined as \( a * b = (a+b) \times (a+1) \times (b+1) \), how would the result change?
5. What happens if you apply the operation to negative numbers, such as \( (-3) * 4 \)?

**Tip:** Always perform operations inside parentheses first when working with nested expressions.

Explore how to solve the nested operation problem to find the value of 4 * (5 * 6). This problem involves understanding and applying the defined operation symbol a * b = (a+b) * (a-1) * (b-1). Follow a detailed step-by-step solution with clear explanations and calculations.

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