How to simplify a diabolical expression involving radicalsA friend and I have been working on this problem for hours - how can the following expression be simplified analytically?It equals 1 2 ,, and we have tried the following to no avail:1. Substitution of x = 5 2. Substitution of x = 2 5 3. Substitution of x = 5 + 5 4. Substitution of x = 5 + 5 5. Manipulations by substituting the golden ratioHere goes: 5 + 2 5 2 + 5 ( 5 + 2 5 ) 4 − 10 + 2 5 8 5 ( 5 + 2 5 ) 4 + 5 ⋅ 5 + 2 5 4 Thanks in advance for any help.

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How to simplify a diabolical expression involving radicalsA friend and I have been working on this problem for hours - how can the following expression be simplified analytically?It equals       1    2    ,,  and we have tried the following to no avail:1. Substitution of   x  =      5  2. Substitution of   x  =  2      5  3. Substitution of   x  =  5  +      5  4. Substitution of   x  =      5    +          5      5. Manipulations by substituting the golden ratioHere goes:                                                        5              +              2                              5                                      2                          +                                            5              (              5              +              2                              5                            )                        4                          −                                            10              +              2                              5                                      8                                                                          5              (              5              +              2                              5                            )                        4                          +        5        ⋅                                            5              +              2                              5                                      4                              Thanks in advance for any help.

Rubi Boyle · Accepted Answer

Hint   Let       a  =      5    +    2          5        ,     b  =      10    +    2          5        .    Show         b    =    (          5        −    1    )    a    ,  ,  so scaling the top and bottom of the fraction by 8 yields                                  4          a          +          2                      5                    a          −                      (                          5                        −            1            )            a                                    10          a          +          2                      5                    a                                                          1                                          1                1                                                                      =              1      2

How to simplify a diabolical expression involving radicals A friend and I have been working on this

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