Erick Wyatt

## Answered question

2022-11-25

Find the number of zeros after decimal point in ${0.2}^{25}$,given that $\mathrm{log}2=0.30101$
My attempt: I found the answer as 17.…
Should we add 1 as 17 is the characteristic or leave it as 17?

### Answer & Explanation

Amir Barajas

Beginner2022-11-26Added 10 answers

You've found that ${0.2}^{25}={10}^{-17.47\dots }$, so ${10}^{-18}<{0.2}^{25}<{10}^{-17},$ that is

so we conclude that there are $17$ zéros after the decimal point.

e3r2a1cakCh7

Beginner2022-11-27Added 2 answers

We have
${\mathrm{log}}_{10}{0.2}^{25}=25\cdot {\mathrm{log}}_{10}\frac{2}{10}=25\cdot \left({\mathrm{log}}_{10}2-1\right)\approx -17.4743$
That gives, by monotinicity of $x↦{10}^{x}$ that
${10}^{-18}<{10}^{-17.4743\dots }={0.2}^{25}<{10}^{-17},$
hence ${0.2}^{25}$ has 17 zeros after the decimal point, no need to add 1.

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