Use a calculator to evaluate the following powers

[jai6638]

Hey... requested my professor to give me a few questions to enable me to improve my skills... am having problems solving them however and was hoping that you could guys could help me:

Q1) Use a calculator to evaluate the following powers. Round the results to five decimal placeS. Each of these powers has a rational exponent. Explain how you can use these powers to define 3^( sqrt of 2 ) which has an irrational exponent.

3^(14/10) = 4.65554
3^(141/100)=4.70697
3^(1414/10000)= 4.72770
3^(14142/10000)= 4.72873
3^(141421/100000)= 4.72878
3^(1414213/1000000)= 4.72880


3^( sqrt of 2 ) = 4.728804

so basically the value of 3^(sqrt of 2 ) comes after 3^(1414213/1000000)... hence, you could find the log of 3^14/10 which is .6679697566, then probably do the following : 3^(.667..+.667...+.667...+.667.. +.667 +.667.. +.667).

EDIT: damn .. what i did above doesn't make sense.. I am soo confused.! i do know that i can somehow use logs by finding the log of one value and then adding the solution multiple times to find the value of 3^ ( sqrt of 2 )

EDIT 2: realized that i posted in wrong forum.. my bad.. shall post in general math forum...

 

[HallsofIvy]

Surely, this is not just an exercise in using a calculator! And this isn't a "word problem" so I'm not sure what you meant by that first sentence. The point of the exercise appears to me to be: You have already defined exponentials for any rational power by a^m/n= (a^m)^1/n= ^n\sqrt{a^m}.
Now, how do you define exponentials for irrational numbers? Every irrational number is the limit of some sequence of rational numbers- that's exactly what you are doing when you say, for example, pi= 3.1415926... 3, 3.1, 3.14, 3.141, 3.14159, 3.141592, 3.1415926,... is a sequence of rational numbers (because they are terminating decimals which could be written as a fraction exactly as you did \sqrt{2})

DEFINING a^x to be the limit of a^{r_n} where r_n is a sequence of numbers converging to x is just defining a^x to be continuous.

You are right: "you could find the log of 3^14/10 which is .6679697566, then probably do the following : 3^(.667..+.667...+.667...+.667.. +.667 +.667.. +.667)."

doesn't make sense. Yes, 3^14/10 is, approximately, 0.66790696... but it makes no sense to talk about 3 to a sum of that. Are you confusing the exponent
14/10= 1.4 with the whole thing: 3^14/10?

 

[jai6638]

thanks much for your help.. appreciate it..