Solution: We use the identity \( p^3 + q^3 = (p + q)^3 - 3pq(p + q) \). First, compute \( pq \) using \( (p + q)^2 = p^2 + 2pq + q^2 \). Substituting, \( 10^2 = 58 + 2pq \) gives \( 100 = 58 + 2pq \), so \( pq = 21 \). Then, \( p^3 + q^3 = 10^3 - 3 \cdot 21 \cdot 10 = 1000 - 630 = 370 \). oxed{370}

["# Mastering Cubic Identities: A Step-by-Step Solution Using ( p^3 + q^3 = (p + q)^3 - 3pq(p + q) )", "Understanding algebraic identities can dramatically simplify complex calculations—especially when working with cubes. One powerful identity is:", "[\np^3 + q^3 = (p + q)^3 - 3pq(p + q)\n]", "This equation allows you to compute the sum of two cubes efficiently, provided you know ( p + q ) and ( pq ). In this article, we’ll walk through a practical example using this identity, demonstrating how to calculate ( p^3 + q^3 ) step by step.", "---", "### Why This Identity Matters", "The standard formula for ( (a + b)^3 ) is:", "[\n(a + b)^3 = a^3 + b^3 + 3ab(a + b)\n]", "Rearranged, it becomes:", "[\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n]", "This simplification is invaluable in algebra, geometry, physics, and even computational math where efficiency and accuracy are crucial.", "---", "### Step-by-step Calculation Example", "Let’s apply the identity using concrete values: assume ( p + q = 10 ) and ( pq = 21 ).", "#### Step 1: Use the identity", "We start with:", "[\np^3 + q^3 = (p + q)^3 - 3pq(p + q)\n]", "#### Step 2: Compute ( (p + q)^3 )", "[\n(10)^3 = 1000\n]", "#### Step 3: Multiply ( 3pq(p + q) )", "[\n3 \cdot 21 \cdot 10 = 630\n]", "#### Step 4: Subtract to find ( p^3 + q^3 )", "[\np^3 + q^3 = 1000 - 630 = 370\n]", "---", "### The Final Answer", "[\n\boxed{370}\n]", "---", "### Real-World Applications", "This identity is not just a mathematical curiosity—it’s widely used:", "- In finance and investment analysis, to compute cumulative returns across time periods.\n- In engineering, for modeling cubic growth or volume relationships.\n- In computer algorithms, optimizing performance by reducing cube computations.\n- In academic testing and competitions, solving algebraic and number-based problems faster.", "---", "### Quick Recap: How to Use the Identity", "1. Identify ( p + q ) and ( pq ).\n2. Plug into ( p^3 + q^3 = (p + q)^3 - 3pq(p + q) ).\n3. Calculate cubes, multiply terms, subtract—the result is your answer.", "---", "### Why This Approach Beats Brute Force", "直接利用立方和公式可避免扩展公式展开，节省计算步骤，减少误差，特别适用于已知部分条件的场景。掌握这一技巧，是提升数学竞赛、工程计算和编程能力的关键の一步。", "---", "Copy this clear, efficient method into your toolkit—whether solving exams, optimizing models, or deepening algebraic insight. The cubic identity ( p^3 + q^3 = (p + q)^3 - 3pq(p + q) ) is not just a formula—it’s a powerful tool for smarter, faster problem-solving.", "---", "Sk illed computation, smarter math. Start using algebraic identities today!"]