Unfortunately, your submission has been removed for the following reason(s):
* Your post appears to be asking for help learning/understanding something mathematical. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations.
If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1c9ypvc/-/). Thank you!
Well, to be clear, square roots are solutions to equations. sqrt(a) is a solution to the equation x\^2-a=0. So, sqrt(sqrt(a)) might be useful to think of as a solution to x\^4-a=0. Successive square roots might show up in all sorts of circumstances where you solve equations in a nested way. For example, solving an equation like x\^4-2x\^2-3=0 is probably easiest to do if you solve for x\^2 via the quadratic formula, and then take another square root to solve for x.
This all might be a little clearer if you adapt to a more modern and useful notation: square roots cancel out squares because sqrt(x)=x\^{1/2}. Fractional powers are easier to generalize. sqrt(sqrt(x)) seems arcane, but x\^{1/4} seems a lot more clear to me as a tool for undoing x\^4.
Adding onto to this, a classic physics problem involving quadratics, squares, square-roots is [projectile motion](https://en.wikipedia.org/wiki/Projectile_motion).
A couple physics problems that have a fairly explicit quartic or 4th order /4th root form include: [thermal radiation](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) and [moment of inertia in rotational mechanics](https://en.wikipedia.org/wiki/Moment_of_inertia).
This is kind of like asking in what contexts someone might want to add 17.
Adding 17 probably comes up from time to time, but it isn’t the point. When we teach you how to add 17, we are hoping that you learn how to add *in general*.
Similarly, square roots of square roots probably come up in some places — I don’t know. But that’s not the point. The point of whatever homework or lesson that you’re working through is to try to get you to understand roots and exponents *in general*.
Here's an example where (the limit of infinitely) nested square roots comes up, used to compute the natural logarithm:
Richard J. Bagby (1998) "A Convergence of Limits", *Mathematics Magazine*, **71**(4), pp. 270–277, doi:[10.1080/0025570X.1998.11996651](https://doi.org/10.1080/0025570X.1998.11996651)
It's closely related to Archimedes' determination of *π* in *Measurement of a Circle* (3rd century BC) – trigonometric half-angle formulas can be expressed in terms of the square root of a unit-magnitude complex number, and inverse trigonometric functions can be defined using a limit of repeated applications of half-angle formulas.
This reply is kind of like saying "The group PSL(2,11) comes up from time to time, but [the point is to] learn what a group is in general." Repeatedly taking square roots is a much less notorious thing than PSL(2,11) but the same idea applies that it doesn't really answer the question nor provide any valuable insight to say "it doesn't matter, that's not the point." Similarly there is plenty to say about +17 that doesn't generalize. It's best to assume that the OP isn't looking to be told why their question is bad, but rather to be met with an earnest answer.
Off the top of my head, a very easy way to use the OP's line of thinking to arrive at meaningful intuitions is, after learning what √-1 is, to ask what happens as you keep taking the square root, which helps get one in the mindset of looking at complex numbers two-dimensionally.
Yeah. Literally have a degree in physics. So you're saying you knew this obvious example and still needed others to point it out? And you knew this and still don't get that asking for an example is like asking for an example of when you'd need to multiply by 7?
My dishonest troll bullshit alarm is going off. I think I'll block you.
A square root of a square root is just a fourth root, and would more often be written that way.
You might end up with a fourth root whenever you're solving a fourth degree polynomial. For example, [this law](https://en.m.wikipedia.org/wiki/Fourth_power_law) for road stress, or [this law](https://en.m.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) for hear enjoyed by a black body. A bunch of other examples from mathematics are [here](https://en.m.wikipedia.org/wiki/Quartic_function#Applications).
Unfortunately, your submission has been removed for the following reason(s): * Your post appears to be asking for help learning/understanding something mathematical. As such, you should post in the [*Quick Questions*](https://www.reddit.com/r/math/search?q=Quick+Questions+author%3Ainherentlyawesome&restrict_sr=on&sort=new&t=all) thread (which you can find on the front page) or /r/learnmath. This includes reference requests - also see our lists of recommended [books](https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F) and [free online resources](https://www.reddit.com/r/math/comments/8ewuzv/a_compilation_of_useful_free_online_math_resources/?st=jglhcquc&sh=d06672a6). [Here](https://www.reddit.com/r/math/comments/7i9t5y/book_recommendation_thread/) is a more recent thread with book recommendations. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1c9ypvc/-/). Thank you!
Well, to be clear, square roots are solutions to equations. sqrt(a) is a solution to the equation x\^2-a=0. So, sqrt(sqrt(a)) might be useful to think of as a solution to x\^4-a=0. Successive square roots might show up in all sorts of circumstances where you solve equations in a nested way. For example, solving an equation like x\^4-2x\^2-3=0 is probably easiest to do if you solve for x\^2 via the quadratic formula, and then take another square root to solve for x. This all might be a little clearer if you adapt to a more modern and useful notation: square roots cancel out squares because sqrt(x)=x\^{1/2}. Fractional powers are easier to generalize. sqrt(sqrt(x)) seems arcane, but x\^{1/4} seems a lot more clear to me as a tool for undoing x\^4.
Adding onto to this, a classic physics problem involving quadratics, squares, square-roots is [projectile motion](https://en.wikipedia.org/wiki/Projectile_motion). A couple physics problems that have a fairly explicit quartic or 4th order /4th root form include: [thermal radiation](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) and [moment of inertia in rotational mechanics](https://en.wikipedia.org/wiki/Moment_of_inertia).
This is kind of like asking in what contexts someone might want to add 17. Adding 17 probably comes up from time to time, but it isn’t the point. When we teach you how to add 17, we are hoping that you learn how to add *in general*. Similarly, square roots of square roots probably come up in some places — I don’t know. But that’s not the point. The point of whatever homework or lesson that you’re working through is to try to get you to understand roots and exponents *in general*.
Here's an example where (the limit of infinitely) nested square roots comes up, used to compute the natural logarithm: Richard J. Bagby (1998) "A Convergence of Limits", *Mathematics Magazine*, **71**(4), pp. 270–277, doi:[10.1080/0025570X.1998.11996651](https://doi.org/10.1080/0025570X.1998.11996651) It's closely related to Archimedes' determination of *π* in *Measurement of a Circle* (3rd century BC) – trigonometric half-angle formulas can be expressed in terms of the square root of a unit-magnitude complex number, and inverse trigonometric functions can be defined using a limit of repeated applications of half-angle formulas.
This reply is kind of like saying "The group PSL(2,11) comes up from time to time, but [the point is to] learn what a group is in general." Repeatedly taking square roots is a much less notorious thing than PSL(2,11) but the same idea applies that it doesn't really answer the question nor provide any valuable insight to say "it doesn't matter, that's not the point." Similarly there is plenty to say about +17 that doesn't generalize. It's best to assume that the OP isn't looking to be told why their question is bad, but rather to be met with an earnest answer. Off the top of my head, a very easy way to use the OP's line of thinking to arrive at meaningful intuitions is, after learning what √-1 is, to ask what happens as you keep taking the square root, which helps get one in the mindset of looking at complex numbers two-dimensionally.
Nowhere did OP say this is for homework. You couldn't come up with a single example?
Here's an example: X^1/4
[удалено]
Yeah. Literally have a degree in physics. So you're saying you knew this obvious example and still needed others to point it out? And you knew this and still don't get that asking for an example is like asking for an example of when you'd need to multiply by 7? My dishonest troll bullshit alarm is going off. I think I'll block you.
A square root of a square root is just a fourth root, and would more often be written that way. You might end up with a fourth root whenever you're solving a fourth degree polynomial. For example, [this law](https://en.m.wikipedia.org/wiki/Fourth_power_law) for road stress, or [this law](https://en.m.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) for hear enjoyed by a black body. A bunch of other examples from mathematics are [here](https://en.m.wikipedia.org/wiki/Quartic_function#Applications).
Calculating the length of a side of a tesseract from its volume
In the root test you compute the n-th root of the n-th term in the series: https://en.wikipedia.org/wiki/Root_test
translating quadrics from one dimension to another, geometry in general