How To Really Solve This Tricky Algebra Problem (XII) - A whiteboard style graphic-illustration showing the following information: x + 1/x = 1; x⁵ + (1/x⁵) = What (??)

Welcome to the twelfth entry in the tricky algebra problem series. This series celebrates the joy of algebra, where the featured problems range from beginner to advanced levels.

Before we start with this entry’s algebra problem, I wish you a happy and mathy new year! A fresh new year calls for a fresh new algebra problem, right? So, I thought I’d kick the year off with a fun (and easy) algebra problem.

To start, you are given the following equation with a single unknown (x):

x + (1/x) = 1

Given this equation, your challenge is to figure out the solution for the following equation:

(x⁵) + (1/x⁵) = ?

Can you solve this problem?

Update Post Publishing

I’d like to credit Andrew Pike for pointing out a mistake that I had in my original problem description. I had limited ‘x’ to the set of all real numbers.

But for the given equation, this is not possible; the solutions are complex numbers. However, the solution to the problem is limited to the set of all real numbers. You may consider this as a hint, if you will.

Spoiler Alert

Beyond this section, I will be explicitly discussing the solution to this problem. So, if you wish to solve this problem on your own, I suggest that you pause reading this essay at this point.

Once you are done with your attempt, you may choose to continue reading the essay and compare our approaches. All the best!

This essay is supported by Generatebg

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Reviewing Ground Fundamentals

The trick to solving this algebra problem is to employ this fundamental algebraic property of exponents:

How To Really Solve This Tricky Algebra Problem (XII) — A whiteboard style graphic-illustration showing the following information: a^(m+n) = (a^m)*(a^n)
Exponent fundamentals — Math illustrated by the author

The proof for this property can be surprisingly more complicated than one imagines. Since such a proof is beyond the scope of this essay, I’ll save it for another day.

For now, it is sufficient that we agree that this fundamental algebraic property of exponents holds. Armed with this knowledge, let us see how we can proceed with solving this problem.

Setting Up the Tricky Algebra Problem

For ease of reference, let us start by labelling the given equation as equation-1:

How To Really Solve This Tricky Algebra Problem (XII) — A whiteboard style graphic-illustration showing the following information: Equation-1: x + (1/x) = 1
Equation-1 — Math illustrated by the author

Now, think for a moment as to how we can apply the fundamental property of exponents we just covered to this equation. As you ponder upon that question, let me square equation-1 on both sides to see what it brings us.

We can expand the left-hand side of the resulting expression using the binomial expansion formula as follows:

How To Really Solve This Tricky Algebra Problem (XII) — A whiteboard style graphic-illustration showing the following information: Equation-1: x + (1/x) = 1; Squaring on both sides: [x + (1/x)]² = ¹²; Expanding using the binomial formula x² + (1/x)² + (2*x*(1/x)) = 1; x² + (1/x)² + 2 = 1; Subtracting 2 from both sides: Equation-2: x² + (1/x²) = −1
Equation-2 — Math illustrated by the author

As you can see, this process has led us one step closer to our solution. Now, can you think of a way to move closer to our solution with the fundamental multiplicative property of exponents in mind?

Well, there are several ways in which we can proceed. Let me demonstrate one of these.

Solving the Problem Further

The simplest approach to proceed is to multiply equation-2 and equation-1 to see what it brings us. Equation-1 has x¹ terms and equation-2 has x² terms. Multiplying x¹ by x² would give us x³. So, we may expect x³ terms in the resulting expression.

How To Really Solve This Tricky Algebra Problem (XII) — A whiteboard style graphic-illustration showing the following information: Equation-2: x² + (1/x)² = −1; Equation-1: x + (1/x) = 1; Multiplying Equation-1 and Equation-2: [x² + (1/x)²]*[x + (1/x)] = (−1)*(1); x³ + x + 1/x + 1/x³ = −1; Substituting x + 1/x = 1: x³ + (1/x³) + 1 = -1; Subtracting 1 from both sides: Equation-3: x³ + (1/x³) = -2
Equation-3 — Math illustrated by the author

As you can see, we are now even closer to our final solution than before. Only one last step remains before we arrive at the final solution. Can you guess what it could be?

The Solution to the Tricky Algebra Problem

Equation-2 has terms with x² and equation-3 has terms with x³. When we multiply x² by x³, we get x⁵. So, the simplest approach would be to continue what has worked for us so far. Let us multiply equation-2 by equation-3 and see if we get closer to our solution:

How To Really Solve This Tricky Algebra Problem (XII) — A whiteboard style graphic-illustration showing the following information: Equation-2: x² + (1/x)² = −1; Equation-3: x³ + (1/x³) = −2; Multiplying equation-2 and equation-3: [x² + (1/x)²]*[x³ + (1/x³)] = (−2)*(−1); x⁵ + 1/x + x + 1/x⁵ = 2; x⁵ + 1/x⁵ + 1 = 2; Subtracting 1 from both sides: Solution: x⁵ + 1/x⁵ = 1
The solution — Math illustrated by the author

There you go. In a bit of a plot twist, the solution to our final expression is the same as our given equation: ‘1’.

I hope you enjoyed solving this simple yet engaging algebra problem. If you thought that this was too easy for you, worry not. We are just getting started with this year and there are bound to be much more challenging problems later.

If solving such algebra problems is right up your alley, then keep an eye on this space for more in the future!

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