MATHS POWERS: How can I figure them out quickly?

Question

For example, how can I figure out 0.01075 to the power of 24 without multiplying everything over and over again to find out the answer?

I have to do an exam soon, and I cannot spend too long on it. Please help!

I cannot take in a calculator

Answer 1

You DO NOT NEED to fugure out 0.01075^24 !Just leave the answer in the form 0.01075^24.

Also, leave all logs (which aren't obvious, eg ln 1 = 0 but log17 = log17), same goes for roots (eg sqrt(144)=12, sqrt(60) = 2sqrt(15), sqrt(2) = sqrt(2) ).

Answer 2

As a few others have indicated, if you can't use a calculator a log table can be useful (and this was the standard method before calculators were around). If you have a sufficiently complete log table this will be fine.

However, I wanted to mention another approach that can be quicker in some cases. When you have or can cast the problem into the form (1 + x)^N, where x is small compared to 1, you can use the first few terms of the binomial expansion for (1 + x)^N to approximate the result.

In your case this isn't so great, since although 0.075 is small compared to 1, the large power means that the first several terms are of comparable size (24 * 0.075 = 1.8). To demonstrate, here are the first few terms:
0.01075 ^ 24 = [(1/100)(1 + 0.075)^24]
= [10^(-2)]^(24) . (1 + 0.075)^24
≈ 10^(-48) . (1 + 24 (0.075) + 24 (23) (0.075)^2 / 2) + 24(23)(22) (0.075)^3 / 3!
= 10^(-48) . [1 + 1.8 + 1.5525 + 0.853875]

For non-calculator work you really want to have sufficient accuracy after one or two steps of this process.

Answer 3

You can do repeated squaring to reduce the number of calculations.

As in x^24 = (x^3)^8 = (x^3)^2^2^2
and x^3 = x^2 *x

so first calculate x^3 with 2 multiplications.
Then find x^24 by squaring three times, which is three more multiplications.

So total 5 multiplications, and you'll have the exact answer.

Though I do agree that this is a silly question to ask without a calculator.

Answer 4

you can do this using log rules

for example

0.01075 to the power 24

take the log of it

log (0.01075 to the power 24)

then use the rule log (x to the power y) = y*log(x)

so

log (0.01075 to the power 24) = 24*log(0.01075)

then reverse the log operation by doing

10 to the power (24*log(0.01075))

and you will get the answer (assuming you are using log to the base 10)

edit:

If you can not use a calculator then the method I would use is to approximate

0.01075 as 0.01 or 10^-2

Then raising to the power 24 you just multiply the -2 by 24

giving 10^-48 as an approximate answer.

I doubt if they would want high accuracy if they do not allow calculators.

Answer 5

You must be joking (multiplying over and over)
It is a good thing you don't have to work out 1.023 to the power 568
On your calculator look for the symbol ^
Type in the number, then ^, then 24 then = or enter depending on the calculator

Answer 6

I hope you are allowed a calculator in this exam. You need a scientific one with the function x to the power y. Then you key in 0.01075 and press the appropriate key. Otherwise it is a rather tedious and long calculation.

Answer 7

Are you sure you cannot bring a calculator or use a spreadsheet? Otherwise this type of question is a complete waste of time. I suppose one way of doing this is to cube the number, then square your answer, square your new answer, and square once again. This will reduce it to only 5 operations.

Answer 8

on a calculator multiply the log of ,01075 by 24 and take the antilog of your answer

Answer 9

use the binomial expansion

1.1x10^-2

= (1+0.1)^24 x 10^-2x24

= 10^-48 x (1+2.4+(24x23x0.1^2)/2!+...)

Most of the terms can just be ignored only the first two can be used.

~3.4x10^-48

(the calculator says: 5.6728740583392883226885281173692e-48 so my answer is close enough

Answer 10

use the power button on your calculator