How to calculate 167^0,2 without a calculator

[kev.thomson96]

I was wondering how that could be done.


I tried 0,2 ---> 1/5, then 167 ^(1/5) , which should lead to 5√167 (fifth root of 167) , but I can't seem to move on from there

 

[HallsofIvy]

0,2= 1/5 so you are asking for the principal fifth root of 167. I note that $2^5= 32$ and $3^5= 243$ so I would next try 2.5. $2.5^5= 97.65625$ (yes, I did that "by hand"!). That's less than 167 so I would try 2.75 next and keep going until I got sufficient accuracy.

 

[George Jones]

I did it using the first two terms of a binomial expansion.

$$167^\frac{1}{5} = \left( 243 - 76 \right) = 243^\frac{1}{5} \left(1 - \frac{76}{243} \right)^\frac{1}{5} \doteq 3 -\frac{1}{5} \frac{76}{81}$$

 

[gmax137]

I was wondering how that could be done.


I tried 0,2 ---> 1/5, then 167 ^(1/5) , which should lead to 5√167 (fifth root of 167) , but I can't seem to move on from there

Are you allowed to use log tables?

 

[kev.thomson96]

I don't think so, but I'd like you to elaborate if you can solve it with log.

 

[HallsofIvy]

If $x= 167^{0,2}$ the $log(x)= 0,2 log(167)$.

So: look up the logarithm of 167 in your log table, multiply by 0,2 then look up the number whose logarithm is that.

 

[Borek]

Note: you will not find logarithm of just 167. log table I have here (base 10) contains logs of numbers between 1 and 10, so you will need to express 167 as 1.67*100 and then log(167) = log(1.67)+2.

Not that it changes the general idea, just makes it a little bit more convoluted.

 

[statdad]

"log table I have here..."

Not sure I could put a finger on a log table if pressed. It reminds me of a conversation I had many years ago (1990-ish) with a historian
Historian: "Do you have a slide rule I can use?"
Me: "No, I haven't had one for many years."
Historian: "I thought every mathematician had one."
Me: "Before you go, do you have any papyrus I could have?"
Historian: "Why would you think we still use that?"

He didn't get my humor.

 

[Mark44]

"log table I have here..."

Not sure I could put a finger on a log table if pressed. It reminds me of a conversation I had many years ago (1990-ish) with a historian
Historian: "Do you have a slide rule I can use?"
Me: "No, I haven't had one for many years."
Historian: "I thought every mathematician had one."
Me: "Before you go, do you have any papyrus I could have?"
Historian: "Why would you think we still use that?"

He didn't get my humor.

I haven't used any of mine for some time, but I still have a few slide rules around.

 

[statdad]

I still have two, but they are in my house, on the same shelves as the old roll film cameras my father had 85 years ago.