MHB How would you do this equation?

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The equation 1.05 = (1 + i)^6 is discussed in terms of solving for the variable i, presumed to represent an interest rate. Participants clarify that only the positive real root is relevant for practical applications. To find all six roots, one suggested rewriting the equation and factoring it as a difference of cubes. The focus remains on determining the roots effectively while considering the context of the variable. The conversation emphasizes the mathematical approach to solving the power equation.
Simon T
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1.05 = (1 + i)^6
 
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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?
 
im not too sure what you mean. Here's a screenshot of the question:

2z8qekn.png
 
Well, what does $i$ represent...an interest rate? If so, then we are only interested in the positive real root.
 
it just says solve the power equation
 
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?
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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