A science fiction writer models the growth of a resource-dependent colony on Mars using the equation \( a + b = 12 \) and \( a^2 + b^2 = 80 \). Find \( a^3 + b^3 \).

["Predicting Mars Colony Growth: How Scientists Use Algebra to Power Resource Planning", "In the red expanses of Mars, where every kilogram of water and watt of energy counts, science fiction authors often ground their colonization stories in real-world mathematical models. One compelling scenario models the growth of a resource-dependent settlement using a system of equations:\n[\na + b = 12 \quad \ ext{and} \quad a^2 + b^2 = 80\n]\nwhere ( a ) and ( b ) represent critical resource variables—such as oxygen supply and solar energy output, respectively. But beyond storytelling, solving this system reveals deeper insights into sustainable Mars development.", "### The Power of Connecting Variables", "Understanding how two quantities relate is fundamental in predicting colony scalability. Given:\n[\na + b = 12 \quad \ ext{and} \quad a^2 + b^2 = 80\n]\nwe aim to compute ( a^3 + b^3 )—a key metric for estimating combined capacity or output across interdependent systems.", "#### Step 1: Use the identity for ( a^3 + b^3 )", "Mathematicians use the elegant identity:\n[\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n]\nWhile we know ( a + b = 12 ), we need the product ( ab ) to proceed.", "#### Step 2: Find ( ab ) from the quadratic identity", "Recall:\n[\n(a + b)^2 = a^2 + 2ab + b^2\n]\nSubstitute known values:\n[\n12^2 = 80 + 2ab \Rightarrow 144 = 80 + 2ab\n]\nSolving for ( ab ):\n[\n2ab = 144 - 80 = 64 \Rightarrow ab = 32\n]", "#### Step 3: Plug into the identity for ( a^3 + b^3 )", "Now substitute into:\n[\na^3 + b^3 = (12)^3 - 3 \cdot 32 \cdot 12 = 1728 - 1152 = 576\n]", "### Why This Matters for Mars Colonization", "For science fiction writers and space planners alike, quantifying combined resource capacity through algebra enables better prediction of settlement limits and growth potential. The equation ( a^3 + b^3 = 576 ) could represent total system resilience over time—critical for long-term survival planning, supply chain design, and colony expansion strategies on Mars.", "### Conclusion", "By modeling Mars’ resource dynamics with equations like ( a + b = 12 ) and ( a^2 + b^2 = 80 ), authors and scientists alike uncover powerful insights. Solving for ( a^3 + b^3 = 576 ) demonstrates how mathematical precision fuels credible, futuristic narratives—and supports real-world strategies for humanity’s next giant leap.", "---", "Keywords: Mars colonization, science fiction math, resource modeling, ( a + b = 12 ), ( a^2 + b^2 = 80 ), ( a^3 + b^3 ), sustainable colonies, space economy, algebra in space science"]