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The question in the image asks you to solve for \( a - b \) and \( a^3 + b^3 \), given the following conditions:

- \( a + b = 10 \)
- \( a^2 + b^2 = 58 \)

### Part (a) - Finding \( a - b \):

We know the identity:
\[
(a + b)^2 = a^2 + b^2 + 2ab
\]

Substitute the known values into this equation:

\[
10^2 = 58 + 2ab
\]
\[
100 = 58 + 2ab
\]
\[
2ab = 100 - 58 = 42
\]
\[
ab = \frac{42}{2} = 21
\]

Now use the identity for the difference of squares:
\[
(a - b)^2 = (a + b)^2 - 4ab
\]

Substitute the values:
\[
(a - b)^2 = 10^2 - 4 \times 21
\]
\[
(a - b)^2 = 100 - 84 = 16
\]
\[
a - b = \sqrt{16} = 4
\]

Thus, \( a - b = 4 \).

### Part (b) - Finding \( a^3 + b^3 \):

We use the identity for the sum of cubes:
\[
a^3 + b^3 = (a + b)((a + b)^2 - 3ab)
\]

Substitute the known values:
\[
a^3 + b^3 = 10((10)^2 - 3 \times 21)
\]
\[
a^3 + b^3 = 10(100 - 63)
\]
\[
a^3 + b^3 = 10 \times 37 = 370
\]

Thus, \( a^3 + b^3 = 370 \).

### Final Answers:
- \( a - b = 4 \)
- \( a^3 + b^3 = 370 \)

Would you like further details or clarification on any of the steps?

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Here are 5 related questions:
1. How do you solve for \( a \) and \( b \) separately from the given equations?
2. What is the geometric interpretation of \( a^2 + b^2 = 58 \) in relation to the points \( (a, b) \)?
3. How can you extend this problem to complex numbers?
4. What is the general formula for \( a^n + b^n \) for any \( n \)?
5. Can the value of \( a - b \) be negative? If so, under what conditions?

**Tip**: Always remember the common algebraic identities like the sum and difference of squares, cubes, etc., as they simplify many problems.

If a + b = 10 and a^2 + b^2 = 58, find: (a) a - b (b) a^3 + b^3

Find the values of a - b and a^3 + b^3 using algebraic identities, given that a + b = 10 and a^2 + b^2 = 58. Step-by-step explanation of using the sum and difference of squares and cubes. Suitable for Grades 9-11.

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