Question

How Can I Find The Value Of (10 ^ -0.1)
(Without Using Calculator)

Answer 1

That's not possible. But 10^-1 is. You move your decimal to the left for negative exponents and you move your decimal to the right for positive exponents. If you had a value of 1 X 10^-2 for example you would move your decimal two places to the left which would equal .01 in decimal format. If you are in fact referring to scientific notation.

Answer 2

10^-.1
= 1/10^.1

So you want 1 divided by the tenth root of 10.

In the days before calculators, we would now have to rely on log tables.
We would have proceeded something like this

let x = 10^(-1/10)
logx = log [10^(-1/10) ]
= (-1/10) log10
= (-1/10)(1)
= -1/10 or -.1
= -1 + .9
we would then find the "antilog of .9 from tables to get 7.9433

the -1 would tell us to move the decimal one place to the left to get
10^ -.1 as .79433

log tables were only valid for positive logs, so if you had a negative, it was necessary to split it up into an integer + a positive decimal

in our -1 + .9
the -1 was called the characteristic, and the .9 is called the mantissa.
The mantissa always had to be positive.
so if we had something like
logx = -3.35 it had to be changed to
log = -4 + .65

You are sooo fortunate to have calculators