Hello, thamathkid1729!
First, we must get in the form
. . This requires some Olympic-level gymnastics.
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Check:
. . . . . close enough!
(1 + x)^(1/3) = 1 + (1/3)x - (1/9)x^2 + (5/81)x^3 - (10/243)x^4 + (22/729)x^5
Use this to get a quick numerical estimate of 70^(1/3).
I started by observing that 70 = 10*(1+6)
so... 70^(1/3) = (10*(1+6))^(1/3) = 10^(1/3) * (1+6)^(1/3)
I then used the binomial expansion of (1 + x)^(1/3), plugging in x = 6
I am getting an answer no where close to the true value of 70^(1/3)