What is a simplified expression for e^2.8 - e^2?

[paradigm]

Problem:
What is the final difference in account values if $100,000 is invested in a) a project with returns of 7% / yr or b) one with returns of 5% / yr. Let the period of investment be 40 years.
This problem is quite simple in theory, and I know the answer should be 100,000(1.07^40 - 1.05^40) = $793446.91. However, this problem is for a logic class in which we are not permitted to use calculators.

Using a couple Taylor expansion approximations -- for the range of values in question, (1+x)^n = e^(nx) -- I get the following:
1.07^40 = e^2.8 and 1.05^40 = e^2.
Basically, all I need to know is how to simplify the expression e^2.8 - e^2 , and then I can simply multiply by $100k to get an accurate approximation.

 

[paradigm]

anyone? :/

 

[Astronuc]

Other than using a slide rule or table, one would have to know the expansion of exp(x) or the binomial coefficients for (1 + x)ⁿ, where n = 40, for this problem.

http://en.wikipedia.org/wiki/Binomial_coefficient

See series (11) here - http://mathworld.wolfram.com/MaclaurinSeries.html

 

[paradigm]

yeah, the binomial expansion was my first thought, but because nx (40 * 0.02) is not <<1 in this case, I would have to take out the series to a large and undetermined number of terms to produce an answer that is even remotely within the ball park.