Evaluating the Algebraic Expression \( (a^3 - b^3 - 3a^2b + 3ab^2) \)

The question asks us to find the value of the expression \( (a^3 - b^3 - 3a^2b + 3ab^2) \) when \( a = 17 \) and \( b = 13 \).

Let's look at the given expression:

\( a^3 - b^3 - 3a^2b + 3ab^2 \)

We can rearrange the terms in this expression:

\( a^3 - 3a^2b + 3ab^2 - b^3 \)

This rearranged form looks very familiar. It is the expansion of a common algebraic identity.

Recognizing the Algebraic Identity

Recall the identity for the cube of a difference:

\( (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \)

Comparing our rearranged expression \( a^3 - 3a^2b + 3ab^2 - b^3 \) with the expansion of \( (x - y)^3 \), we can see that if we replace \( x \) with \( a \) and \( y \) with \( b \), we get exactly the given expression.

Therefore, the expression \( (a^3 - b^3 - 3a^2b + 3ab^2) \) is equal to \( (a - b)^3 \).

Substituting the Given Values

We are given that \( a = 17 \) and \( b = 13 \). Now we need to substitute these values into the simplified expression \( (a - b)^3 \).

First, calculate the value of \( (a - b) \):

\( a - b = 17 - 13 \)

\( a - b = 4 \)

Now, cube this result:

\( (a - b)^3 = (4)^3 \)

Calculating the Final Value

To calculate \( 4^3 \), we multiply 4 by itself three times:

\( 4^3 = 4 \times 4 \times 4 \)

\( 4 \times 4 = 16 \)

\( 16 \times 4 = 64 \)

So, the value of the expression \( (a^3 - b^3 - 3a^2b + 3ab^2) \) when \( a = 17 \) and \( b = 13 \) is 64.

Here is a summary of the steps:

1. Identify the given expression: \( a^3 - b^3 - 3a^2b + 3ab^2 \).
2. Rearrange the terms: \( a^3 - 3a^2b + 3ab^2 - b^3 \).
3. Recognize the algebraic identity: \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \).
4. Equate the expression to the identity: \( (a^3 - b^3 - 3a^2b + 3ab^2) = (a - b)^3 \).
5. Substitute the values \( a = 17 \) and \( b = 13 \).
6. Calculate \( a - b = 17 - 13 = 4 \).
7. Calculate \( (a - b)^3 = 4^3 \).
8. Compute the cube: \( 4^3 = 64 \).

Revision Table: Key Algebraic Identities

Understanding common algebraic identities is crucial for simplifying expressions.

Additional Information: Importance of Identities in Algebra

Algebraic identities are equations that are true for all possible values of the variables involved. They are powerful tools in algebra because they help in:

• Simplifying complex algebraic expressions.
• Solving equations more easily.
• Factoring polynomials.
• Performing calculations quickly, as seen in this problem where calculating \( (17-13)^3 \) is much simpler than calculating \( 17^3 - 13^3 - 3 \times 17^2 \times 13 + 3 \times 17 \times 13^2 \).

Recognizing the structure of an expression and matching it to a known identity is a key skill in algebra. In this problem, identifying the expression as \( (a-b)^3 \) allowed for a straightforward calculation.