exponent question: which is bigger, 99999^99998 or 99998^99999?

[ZMS]

Whole numbers raised to very large exponents obviously result in very large numbers.

(a) Determine the number of digits in 2^{1,000,000,000} (two to the one-billion power). Show all your work.

(b) Describe a method that allows you to find the number of digits in any number aⁿ, where a and n are any integers greater than 1. This may be presented as a formula or as a list of steps.

(c) Which is bigger, 99,999^99,998 or 99,998^99,999? Justify your answer.

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Yes, the answers are obvious for some for a and c, but I have no idea how to explain it and show my work for it.

Thank you!

 

[Deleted member 4993]

Whole numbers raised to very large exponents obviously result in very large numbers.

(a) Determine the number of digits in 2^{1,000,000,000} (two to the one-billion power). Show all your work.

(b) Describe a method that allows you to find the number of digits in any number aⁿ, where a and n are any integers greater than 1. This may be presented as a formula or as a list of steps.

(c) Which is bigger, 99,999^99,998 or 99,998^99,999? Justify your answer.

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Yes, the answers are obvious for some for a and c, but I have no idea how to explain it and show my work for it.

Thank you!

What answers did you get for (a) and (c)? I am assuming you guessed the answer. How did you guess those?

 

[JeffM]

Whole numbers raised to very large exponents obviously result in very large numbers.

(a) Determine the number of digits in 2^{1,000,000,000} (two to the one-billion power). Show all your work.

(b) Describe a method that allows you to find the number of digits in any number aⁿ, where a and n are any integers greater than 1. This may be presented as a formula or as a list of steps.

(c) Which is bigger, 99,999^99,998 or 99,998^99,999? Justify your answer.

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Yes, the answers are obvious for some for a and c, but I have no idea how to explain it and show my work for it.

Thank you!

I am not sure why you say the answers to a and c are obvious. But if they are obvious, I'd like to know what you "saw" that allowed you to answer them without the necessity for computations. I cannot do part a without doing some computations. If you did use computations and they are correct, it will be easy for us to help you develop a rationale for them.

 

[ZMS]

Nvm, the answers aren't obvious. I looked at the questions wrong. I thought it was 2x10^... and 9 x 10^..., Sorry for the early confusion, can you guys show me a way to how to solve this question? Thanks.

 

[Deleted member 4993]

Nvm, the answers aren't obvious. I looked at the questions wrong. I thought it was 2x10^... and 9 x 10^..., Sorry for the early confusion, can you guys show me a way to how to solve this question? Thanks.

Can you calculate

Log₁₀[2^(10^9)]?

 

[ZMS]

I'm pretty sure I figured it out. So, I inputted 100000000 log 2 into the calculator and I got 301029995 (301 million), which is the amount of digits. However, I'm confused as to whether I'd add 1 to the total digits it to include the digit '2' in the total digits. Is that 2 already incorporated into the total digits when I did that log calculation?

Similarly, would I add another 5 digits for c).