The Challenging Algebra Puzzle | Olympiad Prep!

The Challenging Algebra Puzzle | Olympiad Prep!
Gear up for Olympiad success with our deep dive into the challenging algebra puzzle! Unravel complex equations, sharpen your problem-solving skills, and ace the competition. Let's conquer this mathematical challenge together! 🏆🔍 #olympiadpreparation #algebra #mathematics #problemsolving #matholympiad #stemeducation #mathletes
Topics covered:
Algebra Challenge
Math Skills
Expression
Simplifying Expressions
How to simplify expressions
Math Olympiad
Algebra
Cube root of unity
Algebraic identities
Algebraic manipulations
Substitutions
Exponent laws
Quadratic equations
Algebraic Challenging Problem
Math Olympiad Preparation
Math Tutorial
Timestamps:
0:00 Introduction
0:33 Substitution
1:50 Cube root of unity
3:26 Quadratic formula
6:02 Properties of w
7:36 Exponent laws
10:32 Evaluating expression
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Thanks for Watching !!
@infyGyan

[@hardamlimit427]

A) 1

[@NadiehFan]

From
x³ + 1/x³ = 2
we get
(x³)² − 2x³ + 1 = 0
(x³ − 1)² = 0
x³ = 1
so we have
x¹⁶ + 1/x²⁵ = x + 1/x
since 16 ≡ 1 (mod 3) and 25 ≡ 1 (mod 3). So we need to evaluate x + 1/x. To do this we could solve x³ = 1 for x, but it is smarter to evaluate x + 1/x directly. We can write
x³ + 1/x³ = 2
as
(x + 1/x)³ − 3(x + 1/x) = 2
and letting
x + 1/x = t
this gives
t³ − 3t = 2
t³ − 3t − 2 = 0
(t³ − 2³) − 3(t − 2) = 0
(t − 2)(t² + 2t + 4 − 3) = 0
(t − 2)(t + 1)² = 0
t = 2 ⋁ t = −1
Since x¹⁶ + 1/x²⁵ = x + 1/x = t it follows that x¹⁶ + 1/x²⁵ can be either 2 or −1. The correct answer therefore is (d).

[@mohammedsaysrashid3587]

a=1