Question: If $ a + b = 7 $ and $ ab = 10 $, what is the value of $ a^3 + b^3 $?
["Title: How to Calculate $ a^3 + b^3 $ When $ a + b = 7 $ and $ ab = 10 $", "When given the equations $ a + b = 7 $ and $ ab = 10 $, many learners wonder: What is the value of $ a^3 + b^3 $? Solving for $ a^3 + b^3 $ using these values doesn’t require finding $ a $ and $ b $ individually — it’s possible using elegant algebraic identities.", "In this article, we’ll explore how to compute $ a^3 + b^3 $ efficiently, using the identity:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "This formula is derived from expanding $ (a + b)^3 $ and adjusting for the $ 3ab(a + b) $ term. Let’s apply it step by step.", "---", "### Step 1: Plug in the known values", "We are given:\n- $ a + b = 7 $\n- $ ab = 10 $", "Using the identity:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "Substitute:", "$$\na^3 + b^3 = (7)^3 - 3(10)(7)\n$$", "---", "### Step 2: Compute each part", "- $ 7^3 = 343 $\n- $ 3 \ imes 10 \ imes 7 = 210 $", "So,", "$$\na^3 + b^3 = 343 - 210 = 133\n$$", "---", "### Final Answer", "$$\n\boxed{133}\n$$", "---", "### Why This Method Works", "This identity leverages symmetric sums. Even if $ a $ and $ b $ are unknown, their sum and product fully determine $ a^3 + b^3 $ — a powerful result in algebra. Whether solving math problems or preparing for exams, remembering such formulas saves time and reduces errors.", "Keywords: $ a^3 + b^3 $ formula, algebraic identity, solve $ a + b = 7 $, $ ab = 10 $, math problem solving, algebra basics, identity derivation, quick calculation tips", "Meta Description: Learn how to compute $ a^3 + b^3 $ efficiently when $ a + b = 7 $ and $ ab = 10 $ using the identity $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $. Step-by-step solution with full explanation."]
