By Pythagoras: Solving the Equation w² + (3w)² = (10√10)² – A Step-by-Step Breakdown

Mathematics has captivated minds for millennia, and few foundational moments in algebra reflect the timeless elegance of geometry and equations—yet one equation rooted in Pythagorean principles reveals a simple yet powerful learning path. Let’s explore how applying Pythagoras’ insight helps solve w² + (3w)² = (10√10)² and leads to the discovery that w = 10.

Understanding the Context

The Equation: A Pythagorean-style Challenge

At first glance, the equation
w² + (3w)² = (10√10)²
may appear complex, but it sits squarely in the tradition of geometric reasoning—reminiscent of how Pythagoras and his followers modeled relationships using squared values. Let’s unpack it step by step.

Step 1: Expand and Simplify Using Algebraic Identities

(1000)² = w² (w²+25) (w²+400) w = 6.078 | StudyX

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(1000)² = w² (w²+25) (w²+400) w = 6.078 | StudyX
Use theorem Pythagoras to calculate DF. DF² = DE + EF DF² = (7)²
Answered: w²+8w+7 w 1 w²-4w-5 5x2+1 x→+∞o 3x²-x… | bartleby
Answered: From the data, 100 (1 + 0.1 jw) 1-0.0004² + jo.028 TF (W ...
If x = u+v, y = r²+ w² = ²+w then find ou1-dx | Filo

Key Insights

Start by simplifying the left-hand side using algebraic identities:

  • (3w)² = 9w²
  • (10√10)² = 100 × 10 = 1000

So the equation transforms into:
w² + 9w² = 1000

Combine like terms:
10w² = 1000

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oxed{i}Question: What is the smallest positive integer whose cube ends in the digits $888$?
Solution: We seek the smallest positive integer $ n $ such that $ n^3 \equiv 888 \pmod{1000} $. This means we want the last three digits of $ n^3 $ to be $888$. We solve this congruence modulo $1000$. Since $1000 = 8 imes 125$, and $8$ and $125$ are coprime, we can use the Chinese Remainder Theorem by solving modulo $8$ and modulo $125$ separately.
First, modulo $8$:

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Final Thoughts

Step 2: Solve for w²

Divide both sides by 10 to isolate w²:
w² = 100

Step 3: Solve for w

Take the square root of both sides:
w = √100 = 10

Since w represents a length or positive measurement, we accept only the positive root.

Why This Method Reflects Pythagoras’ Legacy

This problem exemplifies geometric thinking made algebraic: just as Pythagoras discovered the deep relationship a² + b² = c² to describe right triangles, today we use squared variables to uncover hidden numerical truths. By simplifying expressions involving w² and (3w)², we apply a fundamental principle of scaling—mirroring how Pythagoras’ theorem scales to real-world dimensions and relationships.