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If a + b + c = 2024 and 1/a + 1/b + 1/c = 1/2024
find 1/a²⁰²⁵ + 1/b²⁰²⁵ +1/c²⁰²⁵
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Just pick one of the variables to be 2024 and the other two to be
any opposite non-zero numbers, say, k and -k
Say a = 2024, b = k, c = -k
Then
2014 + k - k = 2024 and 1/2024 + 1/k + 1/(-k) = 1/2024
Then
1/2024²⁰²⁵ + 1/k²⁰²⁵ - 1/k²⁰²⁵ = 1/2024²⁰²⁵
If you want a decimal approximation and you put it in your
calculator like it is you'll get an overflow because no
ordinary device can handle numbers as large as 2024²⁰²⁵
or as small as 1/2024²⁰²⁵
So here's what you do to get it in scientific notation:
Use base 10 logs:
n = 2024^(-2025)
log(n) = log(2025^(-2025)
log(n) = -2025*log(2024)
log(n) = -2025(3.30621058)
log(n) = -6695.076279
n = 10^(-6695.076279)
n = 10^(-6695)*10^(-0.076279)
n = 10^(-6695)*0.8389208728
Divide the 10^(-6695) by 10 and multiply the 0.8389208728 by 10
n = 10^(-6696)*8.389208728
approximately 8.3892078728... × 10^-6696 in scientific notation.
Edwin
