Question: A science educator designs a workshop where each teacher contributes $ a $ lesson plans and $ b $ activities. If $ a + b = 12 $ and $ a^2 + b^2 = 90 $, find $ a^3 + b^3 $.

["Title: Unlocking Math Power: Solving for $ a^3 + b^3 $ in a Teacher Workshop Scenario", "In the world of education, data-driven planning helps teachers grow and innovate. A recent professional development workshop inspired by real classroom dynamics presents an intriguing math challenge rooted in collaborative curriculum development. Participants contributed structured lesson components, modeled mathematically by variables $ a $ and $ b $, where each teacher develops $ a $ lesson plans and $ b $ hands-on activities.", "Given two key constraints:\n1. $ a + b = 12 $\n2. $ a^2 + b^2 = 90 $", "We are asked to compute $ a^3 + b^3 $, a value with meaningful implications—in this case, reflecting the combined educational impact across multiple dynamic teaching tools.", "---", "### Why This Problem Matters in Education", "This problem isn’t just abstract algebra—it mirrors how educators strategically combine resources. By modeling lesson plan development and activity creation through $ a $ and $ b $, school leaders can forecast total content output. Understanding $ a^3 + b^3 $ introduces precision in evaluating growth potential, supporting better workshop design and resource allocation.", "---", "### Step 1: Use Known Identities to Simplify", "We begin with the identity for the sum of cubes:\n$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$\nWe already know $ a + b = 12 $. To proceed, we need the value of $ ab $, which we can derive using the identity:\n$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$", "Substitute known values:\n$$\n12^2 = 90 + 2ab\n\Rightarrow 144 = 90 + 2ab\n\Rightarrow 2ab = 54\n\Rightarrow ab = 27\n$$", "---", "### Step 2: Plug Into the Sum of Cubes Formula", "Now substitute $ a + b = 12 $ and $ ab = 27 $ into the sum of cubes formula:\n$$\na^3 + b^3 = (12)^3 - 3 \cdot 27 \cdot 12\n= 1728 - 972\n= 756\n$$", "---", "### Final Answer", "$$\n\boxed{756}\n$$", "This elegant result reveals that the total educational impact, captured in $ a^3 + b^3 $, equals 756—a powerful number reflecting the combined contribution of lesson plans and activities designed through thoughtful collaboration.", "For educators and curriculum designers, integrating such mathematical modeling into professional development fosters clearer strategic planning. Like $ a $ and $ b $, every contribution counts—but together, they create a transformative learning outcome worth mathematical precision.", "---", "Keywords: $ a^3 + b^3 $, $ a + b = 12 $, $ a^2 + b^2 = 90 $, math in education, teacher workshop planning, algebra application, active learning design", "Meta Description: Solve $ a^3 + b^3 $ when $ a + b = 12 $ and $ a^2 + b^2 = 90 $. Use algebraic identities to find the precise educational impact value—inspired by collaborative teacher curriculum design."]