Exponents Question

[krishna.daswani]

Hi,

I am currently working on a problem and have looked at the solution but cannot seem to understand part of the explanation. In trying to solve a value for "(1)/(2^n) > 0.01" the answer explanation says:

"(1)/(2^n) > 0.01" is EQUIVALENT to "(2^n) < (100)" -- I don't understand the correlation between the two... how is it decided to move 4 decimal places to the right to reach "100" and why was 4 chosen? Could you please explain the logic behind this?

Thanks,
KD

[jnelson0612]

Hi,

I am currently working on a problem and have looked at the solution but cannot seem to understand part of the explanation. In trying to solve a value for "(1)/(2^n) > 0.01" the answer explanation says:

"(1)/(2^n) > 0.01" is EQUIVALENT to "(2^n) < (100)" -- I don't understand the correlation between the two... how is it decided to move 4 decimal places to the right to reach "100" and why was 4 chosen? Could you please explain the logic behind this?

Thanks,
KD

Hi Krishna,
Think of it this way:
1) 1/(2^n) > 1/100 (note I have converted .01 to 1/100 because I think fractions are easier to deal with)
2) I multiply both sides by 100. I get 100/(2^n) > 1.
3) Cross multiply; I get 100 > (2^n).

Hope this makes more sense!

[krishna.daswani]

This is helpful, thanks!

[RonPurewal]

Hi,

I am currently working on a problem and have looked at the solution but cannot seem to understand part of the explanation. In trying to solve a value for "(1)/(2^n) > 0.01" the answer explanation says:

"(1)/(2^n) > 0.01" is EQUIVALENT to "(2^n) < (100)" -- I don't understand the correlation between the two... how is it decided to move 4 decimal places to the right to reach "100" and why was 4 chosen? Could you please explain the logic behind this?

Thanks,
KD

Hi Krishna,
Think of it this way:
1) 1/(2^n) > 1/100 (note I have converted .01 to 1/100 because I think fractions are easier to deal with)
2) I multiply both sides by 100. I get 100/(2^n) > 1.
3) Cross multiply; I get 100 > (2^n).

Hope this makes more sense!

this approach is of course valid, but a faster approach is to take reciprocals of both sides, thus producing the resulting inequality immediately.

note that taking reciprocals of both sides is something you can only do if the quantities on both sides have the same sign (positive or negative) ... but so is step (3) above. (make sure you know that "cross multiply" is not actually a real operation, and that you can get in serious trouble by using it on some inequalities; what's actually happening when you "cross multiply" is that you are multiplying by the product of both denominators.)

luckily, exponentials must be positive, so we know that both sides of the inequality in this problem are positive, thus rendering either of these approaches possible.