Find the last three digits of the product of the positive roots of:
3√(2021)* Xlog2021X = X3
Wolfram Alpha finds the solutions
x =˜ 2.519407255517746057
x =˜ 3.276427517989203604 *×10^9
After asking WA for more digits, in Matlab:
>> x1=sym(2.519407255517746056766510572536581168853543846177713379873997078370430992634480)
x1 =
5673200788572123 / 2251799813685248
>> x2=sym(3276427517.989203604273896738706379238389746141550490419356385797904605017251962)
x2 =
3435583261103047/1048576
>> double(x1*x2)
ans =
8254655261
>> vpa(x1*x2)
ans =
8254655261.0000002420284485774112
Ignoring rounding error and assuming an integral value,
the last 3 digits are 261.
Later, being more careful to put sym where needed to keep accuracy:
>> x = sym(exp((-1/sym(6) *(sqrt(sym(69)) - 9)* (log(sym(43)) + log(sym(47))))))
x =
exp(-(69^(1/2)/6 - 3/2)*(log(43) + log(47)))
>> y= sym(exp(1/sym(6)* (9 + sqrt(sym(69))) *(log(sym(43)) + log(sym(47)))))
y =
exp((69^(1/2)/6 + 3/2)*(log(43) + log(47)))
>> x*y
ans =
exp((69^(1/2)/6 + 3/2)*(log(43) + log(47)))*exp(-(69^(1/2)/6 - 3/2)*(log(43) + log(47)))
>> vpa(x*y)
ans =
8254655261.0
verifying 261.
Edited on February 20, 2024, 9:35 am
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Posted by Charlie
on 2024-02-20 09:12:15 |
computer-aided solution
