NOAH WILLIAMS
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Thursday 12 March 2026 12:12:24 GMT
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![Stop wasting time with long math steps on your exams! 🤫 Use this 5-second AM-GM formula cheat code to maximize products instantly. Save this video for your next test! 🧠✨ how AM-GM inequality drived How the AM-GM formula works for 3 variables: We use a famous identity for any three positive real numbers x, y, and z: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) We can rewrite the second part of that equation like this: x^2 + y^2 + z^2 - xy - yz - zx = 0.5 * [(x-y)^2 + (y-z)^2 + (z-x)^2] Since squares are never negative, this entire part is greater than or equal to 0: x^3 + y^3 + z^3 - 3xyz >= 0 x^3 + y^3 + z^3 >= 3xyz Now, substitute x with a^(1/3), y with b^(1/3), and z with c^(1/3): a + b + c >= 3 * (abc)^(1/3) Divide both sides by 3 to get the final formula: (a + b + c) / 3 >= (abc)^(1/3) This proves that the Arithmetic Mean is always greater than or equal to the Geometric Mean for 3 variables! #MathHacks #ExamShortcuts #MathTricks #AlgebraHack #SATPrep](/video/cover/7676521844878494997.webp)
