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?

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.