Is √(x²) Always x? The Root of a Square and the Root of a Sum

Is the square root of x² always x? Try −5: √((−5)²) = √25 = 5, not −5. For every real number a, √(a²) is the non-negative one of a and −a — its absolute value |a| — proved case by case on a number line. Then a tool, proved through the factorisation b² − a² = (b − a)(b + a): two non-negative numbers are ordered like their squares, so √7 is compared with 2.7 without a calculator. And the second shortcut falls: for positive a and b, √(a + b) is smaller than √a + √b, because the square of √a + √b carries the extra middle term 2√ab — seen on an area square. Your turn: √(36 + 64) against √36 + √64. Timestamps: 0:00 — Two shortcuts 0:43 — The root of a square 1:11 — Why the absolute value 1:44 — Not minus five 2:02 — Ordered like their squares 2:26 — Why the order is kept 3:02 — √7 or 2.7? 3:26 — The root of a sum 4:17 — Your turn 4:56 — Let's remember 🎓 Free interactive lesson: https://commeunjeu.org/cours/ib-myp-4... ▶️ The whole series “Radicals”:    • Radicals   🎵 Music: "Ever Mindful" by Kevin MacLeod (incompetech.com) Licensed under Creative Commons: By Attribution 4.0 http://creativecommons.org/licenses/b...