Is it even possible to solve 248.83=100*(1+x)^5 for x ?

[Alligator]

Hello everyone,

I'm new to calculus. Could I ask if it is possible to solve this equation for x?

248.83=100*(1+x)^5

(It is a formula from a compounding interest calculator. I am trying to solve for the compounding interest rate, often referred to as the 'effective interest rate' when the Future Value (3200) is known.)

For reference/context:

x is the compounding interest rate
248.83 is the Final Value (ie the balance after 5 years of investment)
100 is the initial investment (ie $100)
1 is the number of compounding periods per year
... and the power of 5 is for 5 years.

As mentioned I don't know if it is even possible to isolate x on one side of the equation but knowing either way will be very helpful.

Thank you very much for your time!

 

[ksdhart2]

Yes, solving for x is definitely possible. Even without doing any calculations, you can immediately tell that there must be exactly five solutions, because if you expand (1 + x)⁵ and gather everything on one side, you'll get a fifth-degree polynomial. The Fundamental Theorem of Algebrahttps://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html says that a fifth-degree polynomial will always have five roots. However, these solutions may or may not be real, and they may not even be unique. In this specific case, however, it turns out there's only one real solution. In any case, to begin solving for ^x, let's first divide both sides by 100:

\(\displaystyle 248.83 = 100 \cdot (1+x)^5 \implies 2.4883 = (1+x)^5\)

Next, it seems as though a review of the Rules/Laws of Exponentshttps://www.rapidtables.com/math/number/exponent.html is in order. From the above, what would happen if you raised both sides to the 1/5-th power (i.e. took the fifth root of both sides)? Where do you think you'd go from there?

 

[haktas]

It is possible to solve for x

Here is how;

Step 1: Divide both sides by 100.Step 2: Take the degree 5 root of both sides of the equatiuon.

Step 3: Subtract 1 from both sides.

Hope this is helpful.