Question: Let $ a $ and $ b $ be complex numbers such that $ a + b = 4 $ and $ a^2 + b^2 = 10 $. Find $ a^3 + b^3 $.
["Title: How to Compute $ a^3 + b^3 $ Given $ a + b = 4 $ and $ a^2 + b^2 = 10 $", "Meta Description:\nDiscover how to find $ a^3 + b^3 $ using the identities of symmetric sums, given $ a + b = 4 $ and $ a^2 + b^2 = 10 $. Step-by-step solution explained.", "---", "Introduction\nWorking with complex numbers can sometimes feel daunting, but applying algebraic identities simplifies the process. A common problem involves finding expressions like $ a^3 + b^3 $, especially when sums and sums of squares of two complex numbers $ a $ and $ b $ are known. In this article, we solve the question:\nLet $ a $ and $ b $ be complex numbers such that $ a + b = 4 $ and $ a^2 + b^2 = 10 $. Find $ a^3 + b^3 $.", "---", "Step 1: Use the identity for $ a^3 + b^3 $\nThere’s a powerful algebraic identity:\n[\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n]\nWe are given $ a + b = 4 $, so we can plug this in:\n[\na^3 + b^3 = 4^3 - 3ab \cdot 4 = 64 - 12ab\n]\nTo compute $ a^3 + b^3 $, we need the value of $ ab $.", "---", "Step 2: Relate $ a^2 + b^2 $ to $ ab $\nWe use the identity:\n[\na^2 + b^2 = (a + b)^2 - 2ab\n]\nSubstitute known values:\n[\n10 = 4^2 - 2ab = 16 - 2ab\n]\nSolve for $ ab $:\n[\n2ab = 16 - 10 = 6 \quad \Rightarrow \quad ab = 3\n]", "---", "Step 3: Plug $ ab $ back into the expression\nNow that $ ab = 3 $, substitute into the earlier expression for $ a^3 + b^3 $:\n[\na^3 + b^3 = 64 - 12 \cdot 3 = 64 - 36 = 28\n]", "---", "Conclusion\nGiven $ a + b = 4 $ and $ a^2 + b^2 = 10 $, we’ve shown that $ a^3 + b^3 = 28 $. This elegant result showcases how symmetric identities simplify complex number computations. Whether $ $a$ $ and $b $ $ are real or complex, algebraic identities remain powerful tools.", "---", "Key Takeaways:\n- Use $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $\n- Express $ ab $ using $ a^2 + b^2 = (a + b)^2 - 2ab $\n- Substituting known values leads directly to the result", "---", "Tagline for SEO:\nMaster complex number identities — compute $ a^3 + b^3 $ using $ a + b $ and $ a^2 + b^2 $ with clear steps and formulas.", "---\nKeywords: complex numbers, $ a^3 + b^3 $ identity, $ a + b = 4 $, $ a^2 + b^2 = 10 $, algebraic computation, sum of cubes formula, complex algebra", "---", "Backend Note for Search Engines:\nOptimized for queries like "calculate $ a^3 + b^3 $ given $ a + b $ and $ a^2 + b^2 $" with precise step-by-step elucidation, ideal for students, educators, and developers working with complex numbers."]
