Expand $sqrt1-x$ up to and including the term $x^2$ (How should i proceed further)

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I am trying to solve a 2 part question, the first part is to expand $sqrt1-x$ up to and including the term $x^2$ which I did.



This gives me $1-(0.5)x-(0.125)x^2 + ...$



However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $sqrt 7 = 10837/4096$. How do i do this part?



EDIT
(After Hint Given By SimpliFire):



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 $$
$$ =frac83sqrt1-frac164;=sqrt7 $$
$$ =frac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right); $$
$$ =frac83left(1-frac1128-frac132768right); $$
$$ =frac83left(frac3276832768-frac25632768-frac132768right); $$
$$ =frac83left(frac3251132768right); $$
$$ =frac26008898304 $$
$$ frac108374096;approxsqrt7 $$



I'm not sure if I can present it like this, if it is okay i will post it as my answer :X






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  • 5




    HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
    – TheSimpliFire
    Sep 1 at 7:58










  • ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
    – deviljones
    Sep 1 at 8:26










  • You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
    – TheSimpliFire
    Sep 1 at 8:27











  • Thanks! I have posted my working, im not sure if the presentation is correct...
    – deviljones
    Sep 1 at 8:51














up vote
2
down vote

favorite












I am trying to solve a 2 part question, the first part is to expand $sqrt1-x$ up to and including the term $x^2$ which I did.



This gives me $1-(0.5)x-(0.125)x^2 + ...$



However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $sqrt 7 = 10837/4096$. How do i do this part?



EDIT
(After Hint Given By SimpliFire):



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 $$
$$ =frac83sqrt1-frac164;=sqrt7 $$
$$ =frac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right); $$
$$ =frac83left(1-frac1128-frac132768right); $$
$$ =frac83left(frac3276832768-frac25632768-frac132768right); $$
$$ =frac83left(frac3251132768right); $$
$$ =frac26008898304 $$
$$ frac108374096;approxsqrt7 $$



I'm not sure if I can present it like this, if it is okay i will post it as my answer :X






binomial-theorem







share|cite|improve this question



















  • 5




    HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
    – TheSimpliFire
    Sep 1 at 7:58










  • ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
    – deviljones
    Sep 1 at 8:26










  • You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
    – TheSimpliFire
    Sep 1 at 8:27











  • Thanks! I have posted my working, im not sure if the presentation is correct...
    – deviljones
    Sep 1 at 8:51












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I am trying to solve a 2 part question, the first part is to expand $sqrt1-x$ up to and including the term $x^2$ which I did.



This gives me $1-(0.5)x-(0.125)x^2 + ...$



However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $sqrt 7 = 10837/4096$. How do i do this part?



EDIT
(After Hint Given By SimpliFire):



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 $$
$$ =frac83sqrt1-frac164;=sqrt7 $$
$$ =frac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right); $$
$$ =frac83left(1-frac1128-frac132768right); $$
$$ =frac83left(frac3276832768-frac25632768-frac132768right); $$
$$ =frac83left(frac3251132768right); $$
$$ =frac26008898304 $$
$$ frac108374096;approxsqrt7 $$



I'm not sure if I can present it like this, if it is okay i will post it as my answer :X






binomial-theorem







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I am trying to solve a 2 part question, the first part is to expand $sqrt1-x$ up to and including the term $x^2$ which I did.



This gives me $1-(0.5)x-(0.125)x^2 + ...$



However, the 2nd part of the question says that by taking $x=(1/64)$ in the expansion of part (i) -> which refers to the one above, deduce that $sqrt 7 = 10837/4096$. How do i do this part?



EDIT
(After Hint Given By SimpliFire):



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 $$
$$ =frac83sqrt1-frac164;=sqrt7 $$
$$ =frac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right); $$
$$ =frac83left(1-frac1128-frac132768right); $$
$$ =frac83left(frac3276832768-frac25632768-frac132768right); $$
$$ =frac83left(frac3251132768right); $$
$$ =frac26008898304 $$
$$ frac108374096;approxsqrt7 $$



I'm not sure if I can present it like this, if it is okay i will post it as my answer :X






binomial-theorem




binomial-theorem






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edited Sep 1 at 8:50

























asked Sep 1 at 7:55









deviljones

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  • 5




    HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
    – TheSimpliFire
    Sep 1 at 7:58










  • ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
    – deviljones
    Sep 1 at 8:26










  • You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
    – TheSimpliFire
    Sep 1 at 8:27











  • Thanks! I have posted my working, im not sure if the presentation is correct...
    – deviljones
    Sep 1 at 8:51












  • 5




    HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
    – TheSimpliFire
    Sep 1 at 7:58










  • ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
    – deviljones
    Sep 1 at 8:26










  • You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
    – TheSimpliFire
    Sep 1 at 8:27











  • Thanks! I have posted my working, im not sure if the presentation is correct...
    – deviljones
    Sep 1 at 8:51







5




5




HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
– TheSimpliFire
Sep 1 at 7:58




HINT: $$sqrt1-frac164=fracsqrt638=frac38sqrt7$$
– TheSimpliFire
Sep 1 at 7:58












ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
– deviljones
Sep 1 at 8:26




ah... i think i solved it, but im not sure if I presented my workings correctly. How do I post my answer here with the nice square root and fractional formattings?
– deviljones
Sep 1 at 8:26












You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
– TheSimpliFire
Sep 1 at 8:27





You could answer your own question. Here is a MathJax tutorial. If you want us to check your solution, then you should instead put it in the original post (above).
– TheSimpliFire
Sep 1 at 8:27













Thanks! I have posted my working, im not sure if the presentation is correct...
– deviljones
Sep 1 at 8:51




Thanks! I have posted my working, im not sure if the presentation is correct...
– deviljones
Sep 1 at 8:51










1 Answer
1






active

oldest

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up vote
1
down vote



accepted










Correct approach!



Some minor notational errors. Here's how I would lay it out:



Since



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 impliesfrac83sqrt1-frac164=sqrt7,$$ from the previous part, we have the approximation
$$beginalignsqrt7approxfrac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right)&=frac83left(1-frac1128-frac132768right) \&=frac83left(frac3276832768-frac25632768-frac132768right)\&=frac83left(frac3251132768right)=frac26008898304=frac108374096endalign$$ as required.






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  • Nicely put, thank you so much :D
    – deviljones
    Sep 1 at 10:45










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










Correct approach!



Some minor notational errors. Here's how I would lay it out:



Since



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 impliesfrac83sqrt1-frac164=sqrt7,$$ from the previous part, we have the approximation
$$beginalignsqrt7approxfrac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right)&=frac83left(1-frac1128-frac132768right) \&=frac83left(frac3276832768-frac25632768-frac132768right)\&=frac83left(frac3251132768right)=frac26008898304=frac108374096endalign$$ as required.






share|cite|improve this answer




















  • Nicely put, thank you so much :D
    – deviljones
    Sep 1 at 10:45














up vote
1
down vote



accepted










Correct approach!



Some minor notational errors. Here's how I would lay it out:



Since



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 impliesfrac83sqrt1-frac164=sqrt7,$$ from the previous part, we have the approximation
$$beginalignsqrt7approxfrac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right)&=frac83left(1-frac1128-frac132768right) \&=frac83left(frac3276832768-frac25632768-frac132768right)\&=frac83left(frac3251132768right)=frac26008898304=frac108374096endalign$$ as required.






share|cite|improve this answer




















  • Nicely put, thank you so much :D
    – deviljones
    Sep 1 at 10:45












up vote
1
down vote



accepted







up vote
1
down vote



accepted






Correct approach!



Some minor notational errors. Here's how I would lay it out:



Since



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 impliesfrac83sqrt1-frac164=sqrt7,$$ from the previous part, we have the approximation
$$beginalignsqrt7approxfrac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right)&=frac83left(1-frac1128-frac132768right) \&=frac83left(frac3276832768-frac25632768-frac132768right)\&=frac83left(frac3251132768right)=frac26008898304=frac108374096endalign$$ as required.






share|cite|improve this answer












Correct approach!



Some minor notational errors. Here's how I would lay it out:



Since



$$ sqrt1-frac164=fracsqrt638=frac38sqrt7 impliesfrac83sqrt1-frac164=sqrt7,$$ from the previous part, we have the approximation
$$beginalignsqrt7approxfrac83left(1-frac12left(frac164right)-frac18left(frac164right)^2right)&=frac83left(1-frac1128-frac132768right) \&=frac83left(frac3276832768-frac25632768-frac132768right)\&=frac83left(frac3251132768right)=frac26008898304=frac108374096endalign$$ as required.







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share|cite|improve this answer










answered Sep 1 at 10:25









TheSimpliFire

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  • Nicely put, thank you so much :D
    – deviljones
    Sep 1 at 10:45
















  • Nicely put, thank you so much :D
    – deviljones
    Sep 1 at 10:45















Nicely put, thank you so much :D
– deviljones
Sep 1 at 10:45




Nicely put, thank you so much :D
– deviljones
Sep 1 at 10:45

















 

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