How to simplify an expression?I have tried to simplify this expression for quite a long...

doturitip9

doturitip9

Answered

2022-07-10

How to simplify an expression?
I have tried to simplify this expression for quite a long time now but I can't find how to do it.
( 1 2 + 4 m 1 m 8 m 3 + 1 : 1 2 m 2 m 2 2 m + 1 ) 4 m + 2 2 m 1 1 1 4 m + 4 m 2
Can someone help me with it?

Answer & Explanation

Shawn Castaneda

Shawn Castaneda

Expert

2022-07-11Added 17 answers

Starting with the original expression:
( 1 2 + 4 m 1 m 8 m 3 + 1 : 1 2 m 2 m 2 2 m + 1 ) 4 m + 2 2 m 1 1 1 4 m + 4 m 2
Factoring where possible and changing the order of some terms:
( 1 2 ( 2 m + 1 ) + m 1 ( 2 m + 1 ) ( 4 m 2 2 m + 1 ) : ( 2 m 1 ) 2 m 2 2 m + 1 ) 2 ( 2 m + 1 ) 2 m 1 1 ( 2 m 1 ) 2
Converting division into multiplication by the reciprocal; distributing the outside fraction:
( 1 2 ( 2 m + 1 ) 2 ( 2 m + 1 ) 2 m 1 + m 1 ( 2 m + 1 ) ( 4 m 2 2 m + 1 ) 2 m 2 2 m + 1 ( 2 m 1 ) 2 ( 2 m + 1 ) 2 m 1 ) 1 ( 2 m 1 ) 2
Some cancellations:
( 1 2 ( 2 m + 1 ) 2 ( 2 m + 1 ) 2 m 1 + m 1 ( 2 m + 1 ) ( 4 m 2 2 m + 1 ) 2 m 2 2 m + 1 ( 2 m 1 ) 2 ( 2 m + 1 ) 2 m 1 ) 1 ( 2 m 1 ) 2
1 2 m 1 2 ( m 1 ) 4 m 2 2 m + 1 2 m 2 2 m + 1 ( 2 m 1 ) 2 1 ( 2 m 1 ) 2
Finding common denominators:
8 m 3 8 m 2 + 4 m 1 ( 4 m 2 2 m + 1 ) ( 2 m 1 ) 2 4 m 3 8 m 2 + 6 m 2 ( 4 m 2 2 m + 1 ) ( 2 m 1 ) 2 4 m 2 2 m + 1 ( 4 m 2 2 m + 1 ) ( 2 m 1 ) 2
Adding/subtracting:
4 m 3 4 m 2 ( 4 m 2 2 m + 1 ) ( 2 m 1 ) 2
Factoring one last time:
4 m 2 ( m 1 ) ( 4 m 2 2 m + 1 ) ( 2 m 1 ) 2

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