Question

Solve this system
\(\large (\sqrt [ 3 ]{ 15x^{ 6 }-9y^{ 4 }+7y^{ 2 }-16x+4 } )^{ 2013 }\\\large+(\sqrt [ 4 ]{ 20y^{ 6 }-8x^{ 2 }+7y^{ 6 }-17y^{ 2 }-2 } )^{ 2014 }\\\large =(\sqrt [ 5 ]{ 5x^{ 6 }+6x^{ 5 }-4y^{ 6 }-3y^{ 3 }-3y-4x+3 } )^{ 2015 }\)
and
\(\large 9(x^{10}-10x^9+50x^8-160x^7+360x^6+719x^4-636x^3+392x^2\\\large-152x+31) (81y^5-270y^4+360y^3-243y^2+84y-11)^2\\\large =x^ 6-6x^ 5+18x^ 4y+6x^ 4-72x^3y+16x^3+108x^2y^2-12x^2\\\large-216xy^2 +144xy-24x+216y^3-21y^2+72y-8\)

Answer 1

My god !! @JungHyunRan it would take me a minute at least to read it. :D

Answer 2

^-^

Answer 3

Well give me some time and I would most probably come up with a solution. I am ill today so cannot be too sure of myself ;)

Answer 4

Just wait...

Answer 5

I am just calling Einstein, may be He can do it... :P

Answer 6

lol @waterineyes Einstein???

Answer 7

Yeah a close friend of @waterineyes ;)

Answer 8

Okay one way is to use some online equation solver :D

Answer 9

When I asked that @JungHyunRan has posted a question on OS, he just cut the call... :P

Answer 10

haha... so funny :P

Answer 11

May be he knows how difficult a question you can ask... :)

Answer 12

What happened @vishweshshrimali5 I think you were typing something... :P

Answer 13

Yeah internet problem. I am planning to have a break up with internet :'(

It does not let me check my mail :P

Answer 14

I am missing @mukushla here... :(

Answer 15

Same here

Answer 16

Well let me concentrate XD :P

Answer 17

this problem doesn't look like @mukushla 's level, it just looks like a pain to process the long equations and bit boring :o

Answer 18

May be collecting like terms may help for second equation

Answer 19

I am really cursing the person who created this problem. Couldn't think of anything short and simple to read and solve ?

Answer 20

\[\large (\sqrt [ 3 ]{ 15x^{ 6 }-9y^{ 4 }+7y^{ 2 }-16x+4 } )^{ 2013 }\\\large+(\sqrt [ 4 ]{ 20y^{ 6 }-8x^{ 2 }+7y^{ 6 }-17y^{ 2 }-2 } )^{ 2014 }\\\large =(\sqrt [ 5 ]{ 5x^{ 6 }+6x^{ 5 }-4y^{ 6 }-3y^{ 3 }-3y-4x+3 } )^{ 2015 }\]

2013 is divisible by 3 !!!

\[\large ({ 15x^{ 6 }-9y^{ 4 }+7y^{ 2 }-16x+4 } )^{ 671 }\\\large+(\sqrt [ 2 ]{ 20y^{ 6 }-8x^{ 2 }+7y^{ 6 }-17y^{ 2 }-2 } )^{ 1007 }\\\large =({ 5x^{ 6 }+6x^{ 5 }-4y^{ 6 }-3y^{ 3 }-3y-4x+3 } )^{ 403 }\]

Answer 21

Now my main purpose would be to think of something for this:

\[\large{\sqrt{20y^6-8x^2+7y^6 - 17y^2 - 2}}\]

Answer 22

@JungHyunRan 20y^6 + 7y^6

are you sure ?

Answer 23

Simplify it a bit other than that I don't think you can do anything lol

Answer 24

I mean such a question creator would not be that lenient on solvers

Answer 25

Can you please recheck the equations ?

Answer 26

I am 100% sure it was something else. I have a lot of experience with such mad mathematicians :'( They don't think in favour of us solvers... Believe me

Answer 27

@ganeshie8 left giving me medal :D

Answer 28

Looks like there is some work spotting perfect square terms.

Answer 29

@vishweshshrimali5 It may be 20y^6 + 7x^6

Answer 30

No no @JungHyunRan I am pretty sure it is something else. I checked using that too but that would not give us the answer.

Answer 31

I can say this because I have solved this problem somewhere but where ??????? God know :(

Answer 32

At that time I had found the answer...

Answer 33

Please please please recheck the question ........

Answer 34

This is question, but it was wrote 20y^6+7y^6 :3

Answer 35

Well if that is the case let me retry :)

Answer 36

Well first let me simplify the second equation :)

My previous two tries have gone in vain

Answer 37

Well I found something similar yet different:

http://diendantoanhoc.net/forum/index.php?/topic/119665-left-sqrt315x6-9y47y2-16x4-right-2013left-sqrt420y6-8x27x2-17y-2-right-2014/

I don't know what is written there but the equation looks different yet little bit problematic as it contains -8x^2+7x^2 which I think is NOT what the person really wanted to ask in the question.

Answer 38

I was not able to get any useful result as the equations are very long and complex. I would leave a comment whenever I find out any result...

@JungHyunRan Till then can you please have this problem checked from the source as there are two different versions of the same question.

Vishwesh

Answer 39

\[(15x^6 - 16x -9y^4 +7y^2 +4)^(671) + (-8x^2 +27y^6 - 17y^2 -2)^(1007/2)= (5x^6 +6x^5 -4x -4y^6 -3y^3 - 3y +3)^(403)\] (671),(1007/2) and (403) are to the power ^

Answer 40

(5x^6 + 6x^5 - 4x - 4y^6 - 3y^3 -3y +3)^(403)

Answer 41

@Mateaus , I too solved up till this step but I don't think it is reasonable to use binomial formula for expanding the expressions :)

Answer 42

Yeah I see, I tried the second one and ended with over 10 expansions

Answer 43

\(\color{blue}{\text{Originally Posted by}}\) @vishweshshrimali5
\[\large (\sqrt [ 3 ]{ 15x^{ 6 }-9y^{ 4 }+7y^{ 2 }-16x+4 } )^{ 2013 }\\\large+(\sqrt [ 4 ]{ 20y^{ 6 }-8x^{ 2 }+7y^{ 6 }-17y^{ 2 }-2 } )^{ 2014 }\\\large =(\sqrt [ 5 ]{ 5x^{ 6 }+6x^{ 5 }-4y^{ 6 }-3y^{ 3 }-3y-4x+3 } )^{ 2015 }\]

2013 is divisible by 3 !!!

\[\large ({ 15x^{ 6 }-9y^{ 4 }+7y^{ 2 }-16x+4 } )^{ 671 }\\\large+(\sqrt [ 2 ]{ 20y^{ 6 }-8x^{ 2 }+7y^{ 6 }-17y^{ 2 }-2 } )^{ 1007 }\\\large =({ 5x^{ 6 }+6x^{ 5 }-4y^{ 6 }-3y^{ 3 }-3y-4x+3 } )^{ 403 }\]
\(\color{blue}{\text{End of Quote}}\)


Though I am thinking of finding out the square root of the second term (one with the power 1007 and a square root

Answer 44

Yeah @Mateaus and Wolfram would not allow me to expand and solve or simplify them.

Answer 45

But I am surprised that a person who created this good problem would be so lenient for us to write 20y^6 + 7y^6 instead of 27y^6... It just doesn't seem correct.

What do you think @Mateaus

Answer 46

hmm what if 7y^6 wasn't mean to be raised to the power of 6 but instead to another.

Answer 47

@vishweshshrimali5 unless the person who created it did it for a reason.

Answer 48

Yup you are correct :)