Square Root of x^9 ≠ x^4.5: Exploring the Difference

[TSN79]

Why is it that the square root of x^9 for some x isn't always the same as x^(4,5) ? I tried to do this with x=12 on my TI-89 and it comes very close, the difference comes after like the 8th decimal or something, but shouldn't it be excactly the same?

 

[Zurtex]

The algorithms for generation:

$$\sqrt{x^9}$$

And:

$$x^{4.5}$$

Will be different for some fixed value of x, as it goes calculators aren't perfect. Back in my further maths class we used to compare the error of our graphical calculators, for calculating exponentials where the results tended to lie around 10^9 Sharp calculators would tend be about 0.1 below the actual answer and Casio ones would be about 0.2 above the actual answer. But these problems often only arise over complex numerical calculations that inherently generate error, seems a bit odd that this would be slightly out.

 

[TSN79]

But in theory these two should be exactly the same right?

 

[Curious3141]

Yes, in theory.

 

[Integral]

Humm... I don't see that happening oh my old HP28c.

Yes, $\sqrt {x^9} = x^{4.5}$. I kinda thought all calculators were about the same, makes me appreciate what I have a bit more. It may be in how many digits are calculated and how many are displayed. I believe HPs calculate a couple more digits then they display. IIRC they calculate 15 and display 12.

 

[arildno]

Humm... I don't see that happening oh my old HP28c.

Yes, $\sqrt {x^9} = x^{4.5}$. I kinda thought all calculators were about the same, makes me appreciate what I have a bit more. It may be in how many digits are calculated and how many are displayed. I believe HPs calculate a couple more digits then they display. IIRC they calculate 15 and display 12.

My Dad still thinks his HP41C is just about the best calculator ever made..

 

[whozum]

Calculators use expansions and Taylor series to compute complex functions. They are usually accurate to the amount of digits they display but not perfect.

Rest assured x^9^(1/2) = x^(4.5). You should be able to prove this easily.

 

[Hurkyl]

Actually, I think calculators use much more clever methods than Taylor series, but enough nitpicking.

Anyways, one description I've heard, which I like, is that the operations you do on a calculator are not the operations you do on paper.

For example, numeric addition is not even an associative operation!

 

[mathwonk]

how many places of accuaracy does your calculator have, 7? What is that 1/10,000,000? So imagine a continuous real line of numbers.

let the choice of unit length be say 10,000,000 miles. then divide it up so that you make a mark every mile. then the only numbers on the number line that appear on your calculator are the ones at the mile markers. I.e. every number between the one mile and 2 mile marker, is invisible to your calculator.


calculators basically have no accuracy at all, in the big picture.

 

[Integral]

My Dad still thinks his HP41C is just about the best calculator ever made..

He may well be right.

 

[TSN79]

Well, here is what my TI-89 returns:

$$\sqrt(12^9)=71831,611091496$$
$$12^{4,5}=71831,611091497$$

Not a difference worth mentioning really, but I still find it strange that it displays a diff at all...
What got me spooked in the first place was that I typed in $$\sqrt(12^9)=12^{4,5}$$ and the TI-89 returned "false"! I was like "what?!?", had I missed the whole concept of square roots?? Thanks to you all, I now know I hadn't...

 

[arildno]

If your calculator works with, say only 1 hidden decimal here, I wouldn't think this very strange.

 

[Data]

Even the newest version of Maple (professional-quality math software) makes errors like that if you don't tell it to increase its precision (strangely enough, Maple 8 did not seem to have these problems).

> evalf(sqrt(12^9));
71831.6110914964791171702546751
> evalf(12^4.5);
71831.6110914964791171702546749

Note that if I increase the precision these digits no longer differ:

> Digits:=50;
Digits := 50
> evalf(sqrt(12^9));
71831.611091496479117170254674931538801852019486025
> evalf(12^4.5);
71831.611091496479117170254674931538801852019486024

but the last ones still disagree!

Stupid computers.

 

[Data]

It is a little smarter than your calculator though:

solve(x^4.5 = sqrt(x^9));
x

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