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 and :
We need to find .
Step 1: Use the identity for
We can use the identity for the sum of cubes:
Step 2: Find
To apply the formula, we need to find . We can use the following identity to relate to :
Substitute the known values and :
Simplifying:
Step 3: Calculate
Now, we substitute and into the identity for :
Substitute the known values:
Simplify:
Thus, the value of is .
Do you want further details or have any questions?
Here are 5 related questions to explore:
- What is the product of and in terms of their cube roots?
- How would the solution change if ?
- Can you express using the same method?
- How do you solve for the individual values of and ?
- What happens if both and 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
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