Solve using identities 75 Cube + - 25 Cube + - 50 cube

We are asked to simplify the expression \( 75^3 + (-25^3) + (-50^3) \) using identities.

Let's solve it step by step:

The expression is:
\[
75^3 + (-25^3) + (-50^3)
\]
This can be rewritten as:
\[
75^3 - 25^3 - 50^3
\]

### Step 1: Use the **sum of cubes** identity

We can apply the sum of cubes formula, which is:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
But since the signs in this problem are not all the same, we cannot directly apply this. Instead, let's approach it by calculating the cubes individually:

### Step 2: Calculate each cube

1. \( 75^3 = 75 \times 75 \times 75 = 421,875 \)
2. \( (-25)^3 = -25 \times -25 \times -25 = -15,625 \)
3. \( (-50)^3 = -50 \times -50 \times -50 = -125,000 \)

### Step 3: Simplify the expression

Now, substitute these values back into the original expression:
\[
75^3 + (-25^3) + (-50^3) = 421,875 - 15,625 - 125,000
\]
\[
= 421,875 - 140,625 = 281,250
\]

### Final Answer:
\[
75^3 + (-25^3) + (-50^3) = 281,250
\]

Would you like further details on how the sum of cubes identity is used in similar problems?

Here are 5 related questions for you to explore:
1. How does the sum of cubes identity apply when all terms are positive?
2. What are some common cube identities for simplifying algebraic expressions?
3. How would this change if the numbers were all negative cubes?
4. Can the difference of cubes identity be used in solving this expression?
5. How can this be applied to real-world problems involving volume?

**Tip:** When simplifying expressions involving cubes, always check whether identities like the sum or difference of cubes can simplify the calculation!

Learn how to simplify the cubic expression 75^3 + (-25^3) + (-50^3) step by step using algebraic identities. This solution covers key algebraic concepts, including the sum and difference of cubes, suitable for students in Grades 9-12.

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