Solution: Use identity $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $. First find $ ab $: $ (a + b)^2 = a^2 + 2ab + b^2 \Rightarrow 49 = 35 + 2ab \Rightarrow ab = 7 $. Then $ a^3 + b^3 = 343 - 3 \cdot 7 \cdot 7 = 343 - 147 = \boxed{196} $.

["# Simplify $ a^3 + b^3 $ Using a Powerful Algebraic Identity – Efficient Computation Without Direct Cubes", "When tackling expressions like $ a^3 + b^3 $, many students and learners resort to memorization. However, there’s a powerful identity that transforms computation into a clear, step-by-step process — and saves time and mental effort. In this article, we explore how to use the identity:\n$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$\nTo calculate $ a^3 + b^3 = 196 $ efficiently, we walk through the derivation, find the product $ ab $, and show exactly how the formula delivers the answer.", "---", "## The Hidden Formula: Why It Works", "We start with the well-known algebraic identity:\n$$\n(a + b)^3 = a^3 + b^3 + 3ab(a + b)\n$$\nRearranging gives:\n$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$\nThis identity elegantly replaces direct cube expansions with simple additions and multiplications — making it ideal for mental math or fast computation.", "---", "## Step-by-step Calculation with Example", "Let’s apply this method using the values from a concrete example to solve\n$$\na^3 + b^3 = \boxed{196}\n$$\nassuming $ a + b = 7 $ and $ a^2 + b^2 = 35 $.", "### Step 1: Find $ ab $ using $ (a + b)^2 $", "We know:\n$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$\nSubstitute known values:\n$$\n7^2 = 35 + 2ab \Rightarrow 49 = 35 + 2ab\n$$\nSolve for $ ab $:\n$$\n2ab = 14 \Rightarrow ab = 7\n$$", "### Step 2: Apply the identity", "Now plug into:\n$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$\nWith $ a + b = 7 $ and $ ab = 7 $:\n$$\na^3 + b^3 = 7^3 - 3 \cdot 7 \cdot 7 = 343 - 3 \cdot 49 = 343 - 147 = 196\n$$", "---", "## Why This Method Is Superior", "- Saves computation steps\n- Reduces risk of algebraic errors\n- Works even when direct cube identities are unfamiliar\n- Works with any real values of $ a $ and $ b $ satisfying $ a + b $ and $ a^2 + b^2 $", "---", "## Final Takeaway", "Using $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $ turns a complex cubic expression into a simple, efficient formula. By first finding $ ab = \frac{(a + b)^2 - (a^2 + b^2)}{2} $, you unlock fast, accurate results — perfect for exams, homework, or quick problem-solving.", "Try it next time you face $ a^3 + b^3 $ — especially when cubes aren’t easily expandable or memorized.", "$$\n\boxed{a^3 + b^3 = 196 \quad \ ext{through efficient identity use}}\n$$", "---", "## Keywords for SEO Optimization\n- $ a^3 + b^3 identity $\n- simplify $ a^3 + b^3 $ without cubes\n- how to compute $ a^3 + b^3 $ using $ a + b $ and $ ab $\n- algebra identity trick $ (a + b)^3 - 3ab(a + b) $\n- step-by-step $ ab $ calculation $ a^2 + b^2 = 35 $\n- mental math $ a^3 + b^3 formula", "---", "optimize learning, algebra tips, cubic identities, math problem solving, smart math tricks, efficient computation, find ab algebra."]