Question

3 x 12^-12 but have to show my work. I cannot figure it out without calculator

Answer 1

9514 1404 393

Answer:

1/2,972,033,482,752

Explanation:

If you must do this with pencil and paper, you can do it in a few stages.

`3*12^(-12)=(3)/(12^(12))=(3)/((12^3)^4)=(3)/((1728^2)^2)=(1)/((1728^2)(576\cdot1728))\\\\=(1)/(2985984\cdot995328)=1/(2\,972\,033\,482\,752)`

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You may already know, for example from the number of cubic inches in a cubic foot, that 12^3 = 1728. This power of 12 can be used to shorten the computation of the 12th power of 12 using the rules of exponents. The long multiplication working at each stage is shown in the attachments. Please note that the final denominator here has 13 digits, which most calculators will not show. A spreadsheet may be able to give you the full precision.

If you need a decimal fraction, not the one shown, then you can compute 1/0.2972...×10^-13 ≈ 3.3646...×10^-13 as shown in the 4th attachment. You should use at least as many digits in the divisor as you need significant digits in the quotient.

This is tedious to do by hand, but is not out of the question. Is there some reason a calculator cannot be used?