Find the value of ##\sqrt[5]{0.00000165}##

[RChristenk]

Homework Statement

Find the value of $\sqrt[5]{0.00000165}$ given $\log165=2.2174839$ and $\log697424=5.8434968$

Relevant Equations

Logarithm rules

$\log x=\log\sqrt[5]{0.00000165}$

$\Rightarrow \log x =\dfrac{1}{5}\log0.00000165=\dfrac{1}{5}(\overline{6}.2174839$

$\Rightarrow \dfrac{1}{5}(\overline{10}+4.2174839) = \overline{2}.8434968$

This is the solution I'm given. I don't understand the last line. First, why is $\overline{6}$ rewritten into $\overline{10}$ and $4.2174839$? Second, I am guessing $\dfrac{1}{5}\cdot \overline{10}$ equals $\overline{2}$. But how do you calculate $\dfrac{1}{5}\cdot 4.2174839$ without resorting to the calculator? This is why I don't get why $\overline{6}$ was rewritten like this because there is still a difficult calculation. Thanks!

 

[BvU]

t how do you calculate $\dfrac{1}{5}\cdot 4.2174839$ without resorting to the calculator?

You divide by 10 (easy enough ) and multiply the result by 2 (not that complicated )

You have $1.65 \times 10^{-6} = 16500 \times 10^{-10}$
log base 10 is $4.217 - 10$
$\sqrt[5] { }$ has log $x-2$ with $x = 4.217/5$ between 0 and 1.
hence the $\overline{2}.8434968$
and with $\log 697424=5.8434968$ you shift 7 places to get $0.0697424$

$\$

 

[Gordianus]

I didn't know that in 2024 the logarithm of a number between 0 and 1 was still reported with a bar above the integer part. I thought that this format belonged to my youth, more than 50 years ago.

 

[BvU]

My youth is equally far back and I never encountered this bar ...

 

[Gordianus]

Back then, I learnt (the hard way) how to use the bar. What's the purpose of that relic?