Question: If $ a + b = 7 $ and $ a^2 + b^2 = 35 $, find $ a^3 + b^3 $.
["# How to Find $ a^3 + b^3 $ Given $ a + b = 7 $ and $ a^2 + b^2 = 35 $", "Understanding how to calculate $ a^3 + b^3 $ using simple algebraic identities is a valuable skill in algebra and problem-solving. In this article, we’ll explore step-by-step how to find $ a^3 + b^3 $ when given the equations:", "$$\na + b = 7\n$$\n$$\na^2 + b^2 = 35\n$$", "## Step 1: Use the Identity for $ a^3 + b^3 $", "Recall the algebraic identity:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "This identity expresses $ a^3 + b^3 $ in terms of $ a + b $ and $ ab $, which makes it ideal when you know both sums and products indirectly.", "### What you have:\n- $ a + b = 7 $\n- $ a^2 + b^2 = 35 $", "You don’t yet know $ ab $, so the next step is to compute $ ab $ using the square of the sum.", "## Step 2: Relate $ a^2 + b^2 $ to $ ab $", "We use the identity:", "$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$", "Substitute the known values:", "$$\n7^2 = 35 + 2ab\n$$", "$$\n49 = 35 + 2ab\n$$", "Subtract 35 from both sides:", "$$\n14 = 2ab\n$$", "Divide by 2:", "$$\nab = 7\n$$", "## Step 3: Plug into the Formula for $ a^3 + b^3 $", "Now recall:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "Substitute $ a + b = 7 $ and $ ab = 7 $:", "$$\na^3 + b^3 = 7^3 - 3 \cdot 7 \cdot 7\n$$", "$$\n= 343 - 3 \cdot 49\n$$", "$$\n= 343 - 147\n$$", "$$\n= 196\n$$", "## Conclusion", "When $ a + b = 7 $ and $ a^2 + b^2 = 35 $, the value of $ a^3 + b^3 $ is:", "$$\n\boxed{196}\n$$", "This method not only provides the solution efficiently but also strengthens foundational algebra skills useful for competitions, homework, or everyday problem-solving. Remember: knowing the identity $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) $ and how to derive unknowns like $ ab $ makes symmetric expressions manageable."]
