- #1
Simon T
- 16
- 0
1.05 = (1 + i)^6
How would you do this equation?
- Context: MHB
- Thread starter Simon T
- Start date
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Discussion Overview
The discussion revolves around solving the equation 1.05 = (1 + i)^6, with a focus on the nature of the variable i and the types of roots being sought. Participants explore different interpretations of the equation, including whether to consider all roots or only specific subsets.
Discussion Character
- Exploratory, Debate/contested, Mathematical reasoning
Main Points Raised
- One participant suggests that if i is treated as a variable rather than the imaginary unit, clarification is needed on whether to find all 6 roots or just specific ones (real or positive real).
- Another participant proposes that if i represents an interest rate, the focus should be on finding the positive real root.
- A different participant indicates that the task is simply to solve the power equation without additional context on the variable i.
- One participant introduces a method to find all 6 roots by rewriting the equation and factoring it as a difference of cubes, inviting others to engage with this approach.
Areas of Agreement / Disagreement
Participants express differing views on the interpretation of the variable i and the types of roots to consider. There is no consensus on how to approach the problem or which roots are relevant.
Contextual Notes
The discussion lacks clarity on the definitions and assumptions regarding the variable i, and the implications of these definitions on the solution process remain unresolved.
- #2
MarkFL
Gold Member
MHB
- 13,284
- 12
Assuming $i$ is a variable, and not the imaginary unit, are you interested in all 6 roots, or just the reals, or just the positive reals?
- #3
Simon T
- 16
- 0
im not too sure what you mean. Here's a screenshot of the question:
- #4
MarkFL
Gold Member
MHB
- 13,284
- 12
Well, what does $i$ represent...an interest rate? If so, then we are only interested in the positive real root.
- #5
Simon T
- 16
- 0
it just says solve the power equation
- #6
MarkFL
Gold Member
MHB
- 13,284
- 12
Okay, well to get all 6 roots, I would let:
$$r=\sqrt[3]{\frac{21}{20}}$$
And write the equation as:
$$\left((1+i)^2\right)^3-r^3=0$$
Now, factor this as a difference of cubes...what do you get?
$$r=\sqrt[3]{\frac{21}{20}}$$
And write the equation as:
$$\left((1+i)^2\right)^3-r^3=0$$
Now, factor this as a difference of cubes...what do you get?
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