Math Problem Statement

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

Solution

We are given the following information about two complex numbers aa and bb:

  • a+b=4a + b = 4
  • a2+b2=6a^2 + b^2 = 6

We need to find a3+b3a^3 + b^3.

Step 1: Use the identity for a3+b3a^3 + b^3

We can use the identity for the sum of cubes:

a3+b3=(a+b)((a+b)23ab)a^3 + b^3 = (a + b)((a + b)^2 - 3ab)

Step 2: Find abab

To apply the formula, we need to find abab. We can use the following identity to relate a2+b2a^2 + b^2 to abab:

a2+b2=(a+b)22aba^2 + b^2 = (a + b)^2 - 2ab

Substitute the known values a+b=4a + b = 4 and a2+b2=6a^2 + b^2 = 6:

6=422ab6 = 4^2 - 2ab

Simplifying:

6=162ab6 = 16 - 2ab

2ab=10ab=52ab = 10 \quad \Rightarrow \quad ab = 5

Step 3: Calculate a3+b3a^3 + b^3

Now, we substitute a+b=4a + b = 4 and ab=5ab = 5 into the identity for a3+b3a^3 + b^3:

a3+b3=(a+b)((a+b)23ab)a^3 + b^3 = (a + b)((a + b)^2 - 3ab)

Substitute the known values:

a3+b3=4(423(5))a^3 + b^3 = 4(4^2 - 3(5))

Simplify:

a3+b3=4(1615)=4×1=4a^3 + b^3 = 4(16 - 15) = 4 \times 1 = 4

Thus, the value of a3+b3a^3 + b^3 is 4\boxed{4}.


Do you want further details or have any questions?

Here are 5 related questions to explore:

  1. What is the product of aa and bb in terms of their cube roots?
  2. How would the solution change if a+b=0a + b = 0?
  3. Can you express a4+b4a^4 + b^4 using the same method?
  4. How do you solve for the individual values of aa and bb?
  5. What happens if both aa and bb are real instead of complex?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Complex Numbers
Polynomial Identities

Formulas

a^3 + b^3 = (a + b)((a + b)^2 - 3ab)
a^2 + b^2 = (a + b)^2 - 2ab

Theorems

Sum of Cubes Identity

Suitable Grade Level

Grades 10-12