Let a and b be complex numbers. If a+b=4 and a^2+b^2=6 then what is a^3+b^3?

We are given the following information about two complex numbers \( a \) and \( b \):

- \( a + b = 4 \)
- \( a^2 + b^2 = 6 \)

We need to find \( a^3 + b^3 \).

### Step 1: Use the identity for \( a^3 + b^3 \)

We can use the identity for the sum of cubes:

\[
a^3 + b^3 = (a + b)((a + b)^2 - 3ab)
\]

### Step 2: Find \( ab \)

To apply the formula, we need to find \( ab \). We can use the following identity to relate \( a^2 + b^2 \) to \( ab \):

\[
a^2 + b^2 = (a + b)^2 - 2ab
\]

Substitute the known values \( a + b = 4 \) and \( a^2 + b^2 = 6 \):

\[
6 = 4^2 - 2ab
\]

Simplifying:

\[
6 = 16 - 2ab
\]

\[
2ab = 10 \quad \Rightarrow \quad ab = 5
\]

### Step 3: Calculate \( a^3 + b^3 \)

Now, we substitute \( a + b = 4 \) and \( ab = 5 \) into the identity for \( a^3 + b^3 \):

\[
a^3 + b^3 = (a + b)((a + b)^2 - 3ab)
\]

Substitute the known values:

\[
a^3 + b^3 = 4(4^2 - 3(5))
\]

Simplify:

\[
a^3 + b^3 = 4(16 - 15) = 4 \times 1 = 4
\]

Thus, the value of \( a^3 + b^3 \) is \( \boxed{4} \).

---

Do you want further details or have any questions?

Here are 5 related questions to explore:

1. What is the product of \( a \) and \( b \) in terms of their cube roots?
2. How would the solution change if \( a + b = 0 \)?
3. Can you express \( a^4 + b^4 \) using the same method?
4. How do you solve for the individual values of \( a \) and \( b \)?
5. What happens if both \( a \) and \( b \) are real instead of complex?

**Tip:** Always check for factorization identities in algebraic problems; they can simplify complex expressions significantly!

Let a and b be complex numbers. If a + b = 4 and a^2 + b^2 = 6, then what is a^3 + b^3?

Explore how to find the value of a^3 + b^3 given the conditions a + b = 4 and a^2 + b^2 = 6. This step-by-step solution covers key algebraic concepts like sum of cubes and complex numbers, making it suitable for students in Grades 10-12.

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