How to calculate 167^0,2 without a calculator

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In summary, the conversation is about finding the principal fifth root of 167 using the binomial expansion and log tables. The suggestion is to use log tables to find the logarithm of 167 and then use the binomial expansion to approximate the fifth root. There is also a humorous anecdote about slide rules and papyrus.
  • #1
kev.thomson96
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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
 
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  • #2
0,2= 1/5 so you are asking for the principal fifth root of 167. I note that [itex]2^5= 32[/itex] and [itex]3^5= 243[/itex] so I would next try 2.5. [itex]2.5^5= 97.65625[/itex] (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.
 
  • #3
I did it using the first two terms of a binomial expansion.

$$167^\frac{1}{5} = \left( 243 - 76 \right) = 243^\frac{1}{5} \left(1 - \frac{76}{243} \right)^\frac{1}{5} \doteq 3 -\frac{1}{5} \frac{76}{81}$$
 
  • #4
Kos Drago said:
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?
 
  • #5
I don't think so, but I'd like you to elaborate if you can solve it with log.
 
  • #6
If [itex]x= 167^{0,2}[/itex] the [itex]log(x)= 0,2 log(167)[/itex].

So: look up the logarithm of 167 in your log table, multiply by 0,2 then look up the number whose logarithm is that.
 
  • #7
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.
 
  • #8
"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.
 
  • #9
statdad said:
"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.
 
  • #10
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.
 

Related to How to calculate 167^0,2 without a calculator

1. What is the process for calculating 167^0.2 without a calculator?

The process for calculating 167^0.2 without a calculator involves using the logarithm function. First, take the logarithm of 167 to the base 10, which is approximately 2.222. Then, multiply this result by 0.2, which gives you 0.444. Finally, raise 10 to the power of 0.444, which gives you the answer of approximately 5.059.

2. Can I use a scientific calculator to calculate 167^0.2?

Yes, you can use a scientific calculator to calculate 167^0.2. Simply enter 167, press the exponent (^) button, and then enter 0.2. The calculator will give you the answer of approximately 5.059.

3. What if I don't have access to a calculator at all?

If you do not have access to a calculator, you can still calculate 167^0.2 using manual methods. One method is to use a logarithm table to find the logarithm of 167 and then use the antilog function to find the final answer. Another method is to use the power of estimation and approximation to get a close answer.

4. Is there a faster way to calculate 167^0.2 without a calculator?

Yes, there is a faster way to calculate 167^0.2 without a calculator. You can use the power rule of logarithms, which states that log(base a)(x^y) = y*log(base a)(x). In this case, you can rewrite 167^0.2 as log(base 167)(167^0.2) = 0.2*log(base 167)(167) = 0.2. This simplifies the calculation and gives you the answer of approximately 5.059.

5. Can I use this method to calculate any number raised to a fractional power without a calculator?

Yes, this method can be used to calculate any number raised to a fractional power without a calculator. The key is to use the power rule of logarithms and then solve for the answer using logarithms and exponentials. However, for complex or large numbers, it may be more efficient to use a calculator.

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