Help with simplification of a rational expression (with fractional powers)Can you please help me see what I don't see yet. Here's a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029): ( a 1 / m − a 1 / n ) 2 ⋅ 4 a ( m + n ) / m n ( a 2 / m − a 2 / n ) ( a m + 1 m + a n + 1 n ) Here's my try at it: ( a 1 / m − a 1 / n ) ( a 1 / m − a 1 / n ) ⋅ 4 a ( 1 / m ) + ( 1 / n ) ( a 1 / m − a 1 / n ) ( a 1 / m + a 1 / n ) ⋅ a ( a 1 / m + a 1 / n ) ...which is ( a 1 / m − a 1 / n ) ⋅ 4 a ( 1 / m ) + ( 1 / n ) a ( a 1 / m + a 1 / n ) 2 Wolfram Alpha's simplify stops here, too. I don't see where to go from here. The final form, according to the book, is this: 1 a ( a 1 / m − a 1 / n ) How did they do it?

Question

Help with simplification of a rational expression (with fractional powers)Can you please help me see what I don&#039;t see yet. Here&#039;s a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029):            (              a                  1                      /                    m                    −              a                  1                      /                    n                            )                  2                    ⋅      4              a                  (          m          +          n          )                      /                    m          n                            (              a                  2                      /                    m                    −              a                  2                      /                    n                    )      (                        a                      m            +            1                          m            +                        a                      n            +            1                          n            )      Here&#039;s my try at it:            (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      ⋅      4              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                            (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      (              a                  1                      /                    m                    +              a                  1                      /                    n                    )      ⋅      a      (              a                  1                      /                    m                    +              a                  1                      /                    n                    )      ...which is            (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      ⋅      4              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                            a      (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2            Wolfram Alpha&#039;s simplify stops here, too. I don&#039;t see where to go from here. The final form, according to the book, is this:      1          a      (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      How did they do it?

Marlee Guerra · Accepted Answer

After checking the older edition of the book, I&#039;m quite sure that the original problem looked like this:            (              a                  1                      /                    m                    −              a                  1                      /                    n                            )                  2                            +            4              a                  (          m          +          n          )                      /                    m          n                            (              a                  2                      /                    m                    −              a                  2                      /                    n                    )      (                        a                      m            +            1                          m            +                        a                      n            +            1                          n            )      Now we have:  (      a          1              /            m        −      a          1              /            n            )          2            +    4      a          (      m      +      n      )              /            m      n        =  (      a          1              /            m        +      a          1              /            n            )          2        (      a          2              /            m        −      a          2              /            n        )  (            a              m        +        1              m    +            a              n        +        1              n    )  =  a  (      a          1              /            m        −      a          1              /            n            )          2        (      a          1              /            m        +      a          1              /            n            )    2  Finally we get:            (              a                  1                      /                    m                    −              a                  1                      /                    n                            )                  2                            +            4              a                  (          m          +          n          )                      /                    m          n                            (              a                  2                      /                    m                    −              a                  2                      /                    n                    )      (                        a                      m            +            1                          m            +                        a                      n            +            1                          n            )        =      1          a      (              a                  1                      /                    m                    −              a                  1                      /                    n                    )

abbracciopj · Accepted Answer

(              a                  1                      /                    m                    −              a                  1                      /                    n                    )      ⋅      4              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                            a      (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2              =              (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      ⋅      4              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                    (              a                  1                      /                    m                    −              a                  1                      /                    n                    )              a      (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2            (              a                  1                      /                    m                    −              a                  1                      /                    n                    )      So one need simplify:              (              a                  1                      /                    m                    −              a                  1                      /                    n                            )        2            ⋅      4              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                            (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2              =            4      (              a                  2                      /                    m                    +              a                  2                      /                    n                    −      2              a                  1                      /                    m          +          1                      /                    n                    )              a                  (          1                      /                    m          )          +          (          1                      /                    n          )                            (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2              =            4      (              a                  3                      /                    m          +          1                      /                    n                    +              a                  3                      /                    n          +          1                      /                    m                    −      2              a                  2                      /                    m          +          2          n                    )              (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2            This is the nearest simplification that I could reach. It cannot be simplified as 1 unless   4  (      a          3              /            m      +      1              /            n        +      a          3              /            n      +      1              /            m        −  2      a          2              /            m      +      2      n        )  =            (              a                  1                      /                    m                    +              a                  1                      /                    n                            )        2

Help with simplification of a rational expression (with fractional powers) Can you please help me

Answered question

Answer & Explanation

New Questions in Pre-Algebra