How to simplify 1/cuberoot(2)

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In summary, Simplifying fractions with radicals in the denominator means rewriting the fractions so that any radical now appears in the numerator.
  • #1
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Homework Statement
Simplify 1/cuberoot(2)
Relevant Equations
NA
I know that 1/sqrt(2) can be multiplied by sqrt(2)/sqrt(2) to give sqrt(2)/2

How can I do something similar with 1/cuberoot(2) ??

What is the trick?
 
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  • #2
barryj said:
Homework Statement:: Simplify 1/cuberoot(2)
Relevant Equations:: NA

I know that 1/sqrt(2) can be multiplied by sqrt(2)/sqrt(2) to give sqrt(2)/2

How can I do something similar with 1/cuberoot(2) ??

What is the trick?
What do you need to multiply ##\displaystyle \root 3 \of {2}## by to get ##2## ?
 
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  • #3
barryj said:
I know that 1/sqrt(2) can be multiplied by sqrt(2)/sqrt(2) to give sqrt(2)/2

How can I do something similar with 1/cuberoot(2) ??
Hi Barry,

Please learn to post math here at PF using LaTeX. See the "LaTeX Guide" link below the Edit window. Thank you.
 
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  • #4
barryj said:
I know that 1/sqrt(2) can be multiplied by sqrt(2)/sqrt(2) to give sqrt(2)/2

What is the trick?
Well, you appear to know the trick already!
And tricks can be repeated, hence the one-trick pony expression. Only this time...
 
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  • #5
Hi @barryj. Hope this isn't regarded as too much of a clue ...

##2= \sqrt 2 ~\times \sqrt 2##
##2= \sqrt [3] 2 ~\times~ ? ~\times~ ?##
 
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  • #6
It’s easier to see if you write it as a fractional exponent

## \frac 1 {2^{\frac 1 3}} ##
 
  • #7
Is there a definition of "simplified" being used in this exercise? To me, ##\frac {1} {\root 3 \of {2}} ##
is "simple" already.
 
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  • #8
gmax137 said:
Is there a definition of "simplified" being used in this exercise? To me, ##\frac {1} {\root 3 \of {2}} ##
is "simple" already.
Ideally we should say 'rationalise the denominator' (i.e. express with no surds in the denominator).

However, it is common usage (here in the UK at least) to interpret ‘simplify’ to include rationalising the denominator (where needed) - even if what you are left with looks more complicated than what you started with!
 
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  • #10
gmax137 said:
Is there a definition of "simplified" being used in this exercise? To me
##\frac {1} {\root 3 \of {2}} ## is "simple" already.

PeroK said:
That's what I think, but some mathematicians disagree
As well as virtually all high school algebra and college precalculus textbooks. As I said in the other thread that @PeroK linked to, the concept of "simplifying" fractions with radicals in the denominator means rewriting the fractions so that any radical now appears in the numerator. No doubt this topic precedes computers and calculators, but I'm sure that its basis is historical. For example, using a table to look up an approximation to ##\sqrt 2##, it's much quicker to calculate ##\frac{\sqrt 2}2 \approx \frac{1.4142}2## than it is to calculate ##\frac 1 {\sqrt 2} \approx \frac 1 {1.4142}##.

Even with modern computers, the computation of the "simplified" version above would possibly occur in fewer machine cycles, in that division by 2 is much less expensive than division by a floating point number.

The textbooks that contain these exercises usually are not concerned with expressions such as ##\frac 1 {\sqrt{2 \pi}}## or any similar expressions with radicands other than integers, to the best of my knowledge.

It seems to me that the length of the other thread was mostly a disagreement about the meaning of "simple." I don't think anyone would complain about the usefulness of being able to work with radicals.
 
  • #11
Well, I learned something today. I have no recollection of being required to present fractions only with integer values in the denominator.

I understand the argument about hand calculating root 2 over two, vs 1 over root 2. But my slide rule has the CI (reciprocal) scale, so...

Not trying to prolong this discussion. I missed the earlier thread, and appreciate the responses clarifying what "simplify" means in this context.
 
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  • #12
gmax137 said:
But my slide rule has the CI (reciprocal) scale, so...
Yeah, mine does, too, but if you want more than 2 or so significant digits of precision, you'd do better by looking up ##\sqrt 2## in a table and then either dividing it by 2 or dividing it into 1.
 
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  • #13
To get back on track, @barryj, do you have enough now to figure this out?

BTW, a little bit of LaTeX goes a long way.
$$\frac 1 {\sqrt[3] 2}$$

Raw LaTeX version: $$\frac 1 {\sqrt[3] 2}$$

For standalone TeX, use two pairs of $ characters. For inline, use two pairs of # characters.
 
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  • #14
Mark44 said:
To get back on track, @barryj, do you have enough now to figure this out?

Apparently not: last seen coincides with post #1 time stamp

1677406347419.png

:cry:

##\ ##
 
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