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We are given two equations:

\[
x^2 + y^2 = 45 \quad \text{and} \quad x - y = 5
\]

We need to find the value of \( x^3 - y^3 \).

### Step 1: Simplify the equations

We can rewrite the expression \( x^3 - y^3 \) using the identity:

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

From the first equation, we already have \( x^2 + y^2 = 45 \). Now, we need to find \( xy \). We can square the second equation:

\[
(x - y)^2 = 25 \quad \text{(since } x - y = 5\text{)}
\]

Expanding this:

\[
x^2 - 2xy + y^2 = 25
\]

Substitute \( x^2 + y^2 = 45 \):

\[
45 - 2xy = 25
\]

Solving for \( 2xy \):

\[
20 = 2xy \quad \Rightarrow \quad xy = 10
\]

### Step 2: Calculate \( x^3 - y^3 \)

Now we substitute into the expression for \( x^3 - y^3 \):

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

We already know \( x - y = 5 \), \( x^2 + y^2 = 45 \), and \( xy = 10 \). Thus:

\[
x^3 - y^3 = 5 \left( 45 + 10 \right) = 5 \times 55 = 275
\]

### Final Answer:

The value of \( x^3 - y^3 \) is \( \boxed{275} \).

---

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:

1. How do you derive the identity for \( x^3 - y^3 \)?
2. What are some other identities for cubes of numbers?
3. Can you solve for \( x \) and \( y \) individually using the given equations?
4. How would you solve this problem if it involved higher powers like \( x^4 - y^4 \)?
5. What happens if \( x - y = 0 \)? How does that affect the calculations?

**Tip:** Always look for algebraic identities to simplify seemingly complex expressions!

If x^2 + y^2 = 45 and x - y = 5, then what is the value of x^3 - y^3?

Learn how to solve the equation x^2 + y^2 = 45 and x - y = 5 to find the value of x^3 - y^3 using cubic identities and quadratic relationships. This step-by-step solution includes key algebraic concepts and formulas, perfect for students in Grades 9-11.

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