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Powers without calulators
By Steve Wisdom on Monday, December 02, 2002 - 02:39 pm:

How do I work out 18.62.6 (18.6 to the power of 2.6) on paper. I can do it on a calculator but can't work out how to do it on paper.

18.62 = 18.6 × 18.6 &
18.63 = 18.6 × 18.6 × 18.6

18.62.6 = 18.6 × 18.6 × (18.60.6) but hpow do you work out what 18.60.6 is.

I am baffled by this I'm afraid. Help.

By Andre Rzym on Monday, December 02, 2002 - 05:17 pm:

You picked a difficult example - something like 9.02.5 would be soluble with pencil and paper using your approach.

Are you familiar with logarithms? If you are (and you 'allow' the use of log tables) then you could do it that way.

Andre

By Matthew Smith on Monday, December 02, 2002 - 06:03 pm:

If you really want to do this without a calculator, or even log tables, it's going to take a very long time to get it to even a few decimal places. One way is to use interval bisection, a 'trial and error' method which you might study at GCSE. A faster way is as follows. It's rather complicated, and I don't know how much algebra you know, but here goes...

You showed in your first post that you need to work out 18.60.6. Let's call that number x. Then we have that

x=18.60.6

x=18.63/5

x5=18.63

Now make a guess as to the value of x. Since 18.6 is between 1 and 32, and 13/5 is 1 and 323/5 is 8, we know that x is between 1 and 8, and we might guess x=4, say. Call this first guess g, so we have g=4.

Next we need to improve the guess, to make it more accurate. Suppose the improved value is g+s, where s is a fairly small number. Then ideally we want

(g+s)5=18.63

Since (g+s)5=(g+s)(g+s)(g+s)(g+s)(g+s)

we can get by multiplying out the brackets

(g+s)5=g5+5g4s+10g3s 2+10g2s3+5gs4+s5

This means that we want to pick an s such that

g5+5g4s+10g3s2+10g2s 3+5gs4+s5=18.63

Now we know that g is fairly near the real value of x, which means that s is going to be smaller than g. Therefore we can ignore the terms which have high powers of s in them, because they're going to be smaller than the terms with high powers of g. This gives us

g5+5g4s=18.63

Solving this equation for s, we get

s=(18.63-g5)/(5g4) [Equation *]

Putting g=4 into this equation, you can work out s.

This isn't going to give us the exact value of s, because we ignored all those terms, but it is going to give us an approximate value of s, so that g+s is a better guess than g was. Once you've worked out g+s, call it g'.

Now that g' is our best guess for x, we can go through all this again to get an even better one, by putting g', instead of g, into equation *. This gives us a new s, which we can add to g' to get an even better guess, g''. We can keep on doing this, and our values of g, g', g'', g''' et c. will get closer and closer to the real value of x.

And because equation * only involves multiplication, subtraction and division, you can do it all using pen and paper.

If you do try this, remember only to work to a certain number of decimal places - say one more than you want your answer to - otherwise your divisions will get longer and longer.