Evaluating the Sum of Cubes (25)³ + (-29)³ + (4)³

We need to find the value of the expression $(25)^3 + (-29)^3 + (4)^3$. This expression is in the form of a sum of cubes, $a^3 + b^3 + c^3$.

Let's identify the terms:

• $a = 25$
• $b = -29$
• $c = 4$

There is a useful algebraic identity for the sum of cubes when the sum of the bases is zero. The identity states that if $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

Let's check if the sum of our bases ($a$, $b$, and $c$) is equal to zero:

Sum $= a + b + c = 25 + (-29) + 4$

Sum $= 25 - 29 + 4$

Sum $= -4 + 4$

Sum $= 0$

Since $a + b + c = 0$, we can apply the identity $a^3 + b^3 + c^3 = 3abc$ to easily calculate the value of the expression $(25)^3 + (-29)^3 + (4)^3$.

Using the identity:

$(25)^3 + (-29)^3 + (4)^3 = 3 \times (25) \times (-29) \times (4)$

Now, we calculate the product:

Value $= 3 \times 25 \times (-29) \times 4$

We can group the numbers to simplify the calculation:

Value $= (3 \times 25) \times (-29) \times 4$

Value $= 75 \times (-29) \times 4$

Value $= 75 \times 4 \times (-29)$

Value $= 300 \times (-29)$

To multiply $300$ by $-29$, we multiply $3$ by $29$ and then multiply by $100$, remembering the negative sign:

$3 \times 29 = 87$

So, $300 \times 29 = 8700$.

Therefore, $300 \times (-29) = -8700$.

The value of the expression $(25)^3 + (-29)^3 + (4)^3$ is $-8700$.

This matches one of the given options.

Revision Table: Key Concepts for Sum of Cubes

Additional Information: Applying Algebraic Identities

Algebraic identities are equations that are true for all possible values of the variables involved. They are powerful tools used to simplify expressions, solve equations, and factor polynomials. The identity used in this problem, $a^3 + b^3 + c^3 = 3abc$ when $a+b+c=0$, is particularly useful for evaluating sums of cubes where direct calculation might be tedious, especially with larger numbers or negative numbers.

Understanding when to apply which identity is crucial. In this case, the structure of the problem $(25)^3 + (-29)^3 + (4)^3$ immediately suggests a sum of cubes involving three terms. Checking if the sum of the bases $(25 + (-29) + 4)$ is zero is the key step to determine if the special identity applies. If the sum were not zero, we would have to evaluate each cube individually and then sum them up, which would be much more work.

For example, calculating $(-29)^3$ directly involves multiplying $-29 \times -29 \times -29$. Knowing the identity allows us to bypass this step and perform simpler multiplications ($3 \times 25 \times -29 \times 4$).