/sci/ - Science & Math » Thread #13327871

77KiB, 900x900, confused.jpg

View Same Google iqdb SauceNAO

Why is (sqrt(x))(sqrt(y)) = sqrt(xy)?

Anonymous Sat 26 Jun 18:21:17 2021 No.13327871 View Reply Original Report

Quoted By: >>13327894 >>13327921 >>13327948 >>13327954

I get why (xy)^n = (x^n)(y^n), but not why (sqrt(x))(sqrt(y)) = sqrt(xy).
If someone could explain that'd be great.

Context: I want to study math but I'm worried that if I don't even understand simple things like mentioned it'll be too hard for me. Also, do things like these get explained in advanced mathematics, because no one ever did so far (I'm in high school, 18 though). In topics like Abstract Algebra perhaps?

Anonymous

Anonymous Sat 26 Jun 2021 18:29:06 No.13327894 Report

Quoted By: >>13327903

>>13327871
you need to disassociate the square root operation from the radical symbol.
nth roots can equally be expressed as
$x^{\frac{1}{n}}$
square roots are then
$x^{\frac{1}{2}}$

Anonymous

Anonymous Sat 26 Jun 2021 18:32:16 No.13327903 Report

Quoted By: >>13327921 >>13327922

>>13327894
I know this, but it still doesn't make it clearer. I understand why (xy)^n = (x^n)(y^n), because (xy)^n = xxx..xyy...y = (x^n)(y^n). But this can't be done with fractional exponents

Anonymous

Anonymous Sat 26 Jun 2021 18:39:11 No.13327921 Report

Quoted By: >>13327934 >>13327936

>>13327871
>(I'm in high school, 18 though)
just saw this. In general, most math professors (or at least the ones I have had) explain things better than they were explained to me in college.

I'm not a math major, but I'm doing Electrical Engineering - Signal Processing which is about as close to a non-math math major you can get, outside of physics. I started college being unable to do highschool algebra 1. I had to start from the very beginning, but working up from basic principles made it all click and now I do math all day for fun in my free time.

>>13327903
Ah, I see what you mean now. There comes a point where simple operations take a jump in abstraction. Same thing happens with the factorial - how do we continue it for all positive integers? Etc.

Generalizing the roots and powers comes with a new level of abstraction where we're forced to stops seeing them as iterated multiplication, and instead see them in terms of exponentiation and logarithms. https://math.stackexchange.com/questions/1992464/how-is-irrational-exponent-defined has a pretty good discussion of it.

Anonymous

Anonymous Sat 26 Jun 2021 18:39:22 No.13327922 Report

Quoted By:

>>13327903
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:exp/x2ec2f6f830c9fb89:rational-exp/v/basic-fractional-exponents

this video can give an intuition for it

Anonymous

Anonymous Sat 26 Jun 2021 18:43:10 No.13327934 Report

Quoted By:

>>13327921
>I had to start from the very beginning, but working up from basic principles made it all click and now I do math all day for fun in my free time.
I think I am in the same route to that. Had a really poor background and now building a good basis for exams at least.

Anonymous

Anonymous Sat 26 Jun 2021 18:43:42 No.13327936 Report

Quoted By: >>13327941 >>13328086

>>13327921
>just saw this. In general, most math professors (or at least the ones I have had) explain things better than they were explained to me in college.
But in college these things won't be discussed specifically anymore no? So even if the professors are better, it won't help me understand the kind of things I'm asking about.

>Also, do things like these get explained in advanced mathematics, because no one ever did so far (I'm in high school, 18 though). In topics like Abstract Algebra perhaps?
Do you happen to know something about this?

Anonymous

Anonymous Sat 26 Jun 2021 18:44:44 No.13327941 Report

Quoted By:

>>13327936
Forgot to mention, I'm reading some books to improve my foundations as well, but none explain the things I'm asking about. Pretty frustrating I gotta say

Anonymous

Anonymous Sat 26 Jun 2021 18:45:43 No.13327948 Report

Quoted By: >>13327957 >>13328054

>>13327871
Square both sides, you get the same result.

Anonymous

Anonymous Sat 26 Jun 2021 18:47:29 No.13327954 Report

Quoted By:

>>13327871
Sqrt(x) produces a number with half the logarithm of x (in any base).
You can think of logarithm as measuring the size of a number in terms of how much it multiplies instead of how much it adds.
Cutting two things in half and putting them together is the same as putting them together than cutting in half.
Therefore sqrt() distributes over multiplication.

Anonymous

Anonymous Sat 26 Jun 2021 18:48:03 No.13327957 Report

Quoted By: >>13327982

>>13327948
Doesn't explain it intuitively though

Anonymous

Anonymous Sat 26 Jun 2021 18:52:28 No.13327975 Report

Quoted By:

sqrt(x) is the positive square root of x, which is unique. So by definition, sqrt(x)sqrt(y) is positive and also obviously a square root of xy.

Anonymous

Anonymous Sat 26 Jun 2021 18:54:42 No.13327982 Report

Quoted By: >>13328003

>>13327957
that khan academy video explains it more intuitively

and a lot of maths in college ends up becoming using definitions and rules that have been proven to be correct without you needing to intuitively understand/visualise it.

Anonymous

Anonymous Sat 26 Jun 2021 18:55:08 No.13327986 Report

Quoted By: >>13328006

Suppose
a= sqrt(x)
b=sqrt(y)
then a^2 = x, ,b^2 = y,
(ab)^2 = a^2 b^2 = xy
so ab = sqrt(xy), hence substituting back in
sqrt(x)sqrt(y)=sqrt(xy)

Anonymous

Anonymous Sat 26 Jun 2021 18:59:43 No.13328003 Report

Quoted By: >>13328054

>>13327982
>without you needing to intuitively understand/visualise it.
I want to though

Anonymous

Anonymous Sat 26 Jun 2021 19:00:54 No.13328006 Report

Quoted By:

>>13327986
It says this in my book too (Lang's Basic Mathematics), I'm trying to understand it more intuitively though

Anonymous

Anonymous Sat 26 Jun 2021 19:13:44 No.13328054 Report

Quoted By: >>13328084

>>13328003
>I want to though
you won't be able to with all things though, e.g. try and visualise a 5 dimensional surface.

you will eventually need to rely on definitions and theorem and rules that have been proven.

e.g. >>13327948 is a really elegant way of using the definition of an equation and the rules surrounding it to show why (sqrt(x))(sqrt(y)) = sqrt(xy)

and if you go further into maths, being able to prove things this way becomes a really important skill

e.g.
https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irrational-numbers/x2f8bb11595b61c86:proofs-concerning-irrational-numbers/v/proof-that-square-root-of-2-is-irrational

Anonymous

Anonymous Sat 26 Jun 2021 19:22:05 No.13328084 Report

Quoted By:

>>13328054
>you won't be able to with all things though, e.g. try and visualise a 5 dimensional surface.
I want to as much as possible then

Anonymous

Anonymous Sat 26 Jun 2021 19:22:57 No.13328086 Report

Quoted By: >>13328170

>>13327936
>won't be discussed specifically anymore no?
depends on what you mean by "specifically"
By and large, the general math courses you have to take before you start your major-specific math courses are going to be tailored towards practicality. The average engineer, doctor, programmer (etc.) don't care or need to know the rigorous proof of, or intuition behind the techniques they're learning. They just care about being able to apply the techniques to reach an answer.

For me, once I got to Calculus, it all started being intuitive and it got a lot more interesting. It was all still tailored for practicality, but my professors were definitely getting into their element and starting to show "this is what I enjoy teaching". They really tried to drive home the intuition behind the concepts.

Even then though, there's only so far they can go. They can't explain the intuition behind *Everything*, but eventually (especially once you get to Calculus), you start developing mathematical maturity and the intuitions just start revealing themselves as you go along, without anybody having to show them to you. Of course, you do still need to think things through before the intuition becomes apparent.

Once you get to upper-level/math major classes, you start again. From square one. With your first Analysis class, you learn to define the basic math operations, from basic axioms and you work your way up. I've never done these classes so I can't say much on if they'll help you with your predicament. But, having read and worked through books on Analysis for fun, it definitely seems like you'll be a lot happier once you get to this level.

You're going to hate me saying this (and I hate saying it too, it feels like a cop-out) but some things really just don't have a good intuition. Those things require you to accept the properties as axioms, and build intuition from those instead. I can't say how often topics like that come up in a Math track, but it's rather common in EE.

Anonymous

Anonymous Sat 26 Jun 2021 19:47:41 No.13328170 Report

Quoted By:

>>13328086
>Once you get to upper-level/math major classes, you start again. From square one. With your first Analysis class, you learn to define the basic math operations, from basic axioms and you work your way up. I've never done these classes so I can't say much on if they'll help you with your predicament. But, having read and worked through books on Analysis for fun, it definitely seems like you'll be a lot happier once you get to this level.
This is what I was hoping to hear, thanks

Anonymous

Anonymous Sat 26 Jun 2021 20:03:23 No.13328237 Report

Quoted By:

I feel like this is something you could prove with the binomial theorem

Capcode	All Only User Posts Only Moderator Posts Only Admin Posts Only Developer Posts
Show Posts	All Only With Images Only Without Images
Deleted Posts	All Only Deleted Posts Only Non-Deleted Posts
Ghost Posts	All Only Ghost Posts Only Non-Ghost Posts
Post Type	All Only Sticky Threads Only Opening Posts Only Reply Posts
Results	All Grouped By Threads
Order	Latest Posts First Oldest Posts First

Your latest searches