How Can I Find The Value Of (10 ^ -0.1)
(Without Using Calculator)
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.
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
To find the value of (10^-0.1) without using a calculator, you can use logarithms. Here's the step-by-step process:
1. Recall the logarithm property: log[a^b] = b * log[a].
2. Take the logarithm of both sides of the equation to get:
log[(10^-0.1)] = -0.1 * log[10].
3. Use the fact that log[10] = 1. The equation then becomes:
log[(10^-0.1)] = -0.1 * 1.
4. Simplify the right side of the equation:
log[(10^-0.1)] = -0.1.
5. Now, rewrite the logarithmic equation in exponential form:
10^(log[(10^-0.1)]) = 10^(-0.1).
6. Apply the logarithm property:
10^-0.1 = 10^(-0.1).
Therefore, the value of (10^-0.1) is equal to 10^(-0.1), which is approximately 0.79433.