Question

Hey guys,
See, 2nd root of 4 is Square root of 4, i.e, 2
3rd root of 27 is cube root of 27 , i.e., 3
Similarly, 10th root of 1024 is 2;

Now, The question for Calculator users, what is :- -------> 10^9th root of 2 or say:- What is 10^100th root of 'X' ?
[ Give proper Explanation ] --> [ HINT :--> Use Limits to explain ur answer ]

Answer 1

The questions are not very clear to me.  For very large powers and roots we can do like this.

Q1)   Let  y be a small real number.  y << 1.

       10⁹ th root of 2 = 1 + y  (say).
       1000,000,000 th root of 2 = 1 + y
       (1 + y)¹⁰⁰⁰⁰⁰⁰⁰⁰⁰ = 2
As y << 1,
        1 + 1000,000,000 y = 2
        y = 10⁻⁹       
Answer :    $\sqrt[1000000000]{2}=1.000000001$

Q2:  Let y be a small real value.
          10¹⁰⁰ th root of X = 1 + y
So   $(1 + y)^{10^{100}}=X\\\\1+10^{100}*y=X \: (approx)\\\\y=\frac{X-1}{10^{100}}=(X-1)*10^{-100}\\\\Answer: 1+(X-1)*10^{-100}$