Question

@SolomonZelman will you please help me

Answer 1

if I can

Answer 2

yes wait

Answer 3

If
\[x=3-\sqrt{8}\]
, then
\[x ^{3}+\frac{ 1 }{ x ^{3} }=\]

Answer 4

for the "then" part, just plug in `3-√8` for x.

Answer 5

won't it be tedious

Answer 6

Yes, or at fist it would be...

\(\LARGE\color{black}{ (3-\sqrt{8})^3+ \frac{1}{(3-\sqrt{8})^3} }\)

Answer 7

I was thinking of
(a+b)^3 identity

Answer 8

\(\Large\color{black}{ (3-\sqrt{8})^3+ \frac{1}{(3-\sqrt{8})^3} }\)


use (a-b)³=(a-b)(a²-2ab+b²)

\(\Large\color{black}{ (3-\sqrt{8})(9-2\sqrt{8}+8)+ \frac{1}{(3-\sqrt{8})(9-2\sqrt{8}+8)^3} }\)

Answer 9

that becomes a hell lot complicated

Answer 10

\(\Large\color{black}{ (3-\sqrt{8})(9-2\sqrt{8}+8)+ \frac{1}{(3-\sqrt{8})(9-2\sqrt{8}+8)} }\)

\(\Large\color{black}{ (3-\sqrt{8})(17-2\sqrt{8})+ \frac{1}{(3-\sqrt{8})(17-2\sqrt{8})} }\)

Answer 11

Yes... but we can do something about it....

Answer 12

we instead should use (a+b)^3

Answer 13

I am using (a-b)³ ... just go over my replies... I would have been done by now, if I didn't disconnect 3 times.

Answer 14

\(\large\color{black}{ (51-6\sqrt{8}-17\sqrt{8}+2(8)~)+ \frac{1}{(51-6\sqrt{8}-17\sqrt{8}+2(8)~)} }\)

\(\large\color{black}{ (51-23\sqrt{8}+16~)+ \frac{1}{(51-23\sqrt{8}+26~)} }\)

Answer 15

\(\large\color{black}{ (51-23\sqrt{8}+16~)+ \frac{1}{(51-23\sqrt{8}+26~)} }\)

\(\large\color{black}{ \frac{(51-23\sqrt{8}+26~)^2}{(51-23\sqrt{8}+26~)} + \frac{1}{(51-23\sqrt{8}+26~)} }\)

Answer 16

\(\large\color{black}{ \frac{(51-23\sqrt{8}+26~)^2+1}{(51-23\sqrt{8}+26~)} }\)

Answer 17

you can play around more, expand it and add 1.

Answer 18

MAybe there is an easier way for this

Answer 19

this is quite easy

Answer 20

a cunning way i meant

Answer 21

\(\large\color{black}{ \frac{(77-23\sqrt{8})^2+1}{(77-23\sqrt{8})} }\)

Answer 22

I added inside the parenthesis, forgot to do that.

Answer 23

\(\large\color{black}{ \frac{(77-23\sqrt{8})^2+1}{(77-23\sqrt{8})} }\)


\(\large\color{black}{ \frac{5929-46\sqrt{8}+4232}{77-23\sqrt{8}} }\)


\(\large\color{black}{ \frac{10161-46\sqrt{8}}{77-23\sqrt{8}} }\)

Answer 24

then use a conjunage

\(\large\color{black}{ \frac{10161-46\sqrt{8}}{77-23\sqrt{8}} }\) multiply top and bottom times `77+23√8`

Answer 25

\(\large\color{black}{ \frac{(10161-46\sqrt{8})(77+23\sqrt{8})}{5906} }\)

Answer 26

\(\large\color{black}{ \frac{782397-8464++230161\sqrt{8})}{5906} }\)

Answer 27

\(\Large\color{black}{ \frac{773933+230161\sqrt{8})}{5906} }\)

Answer 28

\[a^3- b^3 = (a-b)^3+ 3ab(a-b)\]

Answer 29

\[\frac{1}{3- \sqrt8} = \frac{3+ \sqrt8}{1}\]

Answer 30

where do you get that, may I ask ?

Answer 31

@Mashy it helped
@SolomonZelman thank you

Answer 32

(a+b)^3= a^3 + b^3 +3ab(a+b)
(a+b)^3-3ab(a+b) = a^3 + b^3


It was so simple lol