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4.`*^-12 Sqrt[3.0254210421890627`*^22 - 6.25`*^22 T6^2]

The above expression is equal to

Sqrt[0.48406736675024997` - T6^2]

How can I tell Mathematica to simplify the first form into the second form?

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3  
Try FullSimplify[MapAt[Factor, yourExpression, {2, 1}]]. –  Silvia Jan 17 '13 at 7:40
    
@Silvia. Actually FullSimplify isn't needed; Simplify will do. –  m_goldberg Jan 17 '13 at 7:54
    
@m_goldberg Yes indeed. It's just my personal habit to use FullSimplify. –  Silvia Jan 17 '13 at 10:15
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1 Answer

Building on Silvia's comment:

expr = 4.`*^-12 Sqrt[3.0254210421890627`*^22 - 6.25`*^22 T6^2];

Rationalize[Factor //@ expr, 0] // Simplify
Sqrt[1936269467001/4000000000000 - T6^2]
% // N
Sqrt[0.484067 - 1. T6^2]

The 1. is an artifact of // N due to - T6^2 being internally Times[-1, Power[T6, 2]].

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1  
Note: with V9, Rationalize isn't needed: Factor //@ expr // Simplify will do. –  m_goldberg Jan 17 '13 at 8:16
    
This does not work for 2. Sqrt[0.25 - b/3] . I just want all things are within Sqrt. –  goodluck Jan 17 '13 at 8:38
    
@goodluck try: Rationalize[2. Sqrt[0.25 - b/3], 0] // FullSimplify –  Mr.Wizard Jan 17 '13 at 8:46
    
@Mr.Wizard, it works for 0.25, but not work for 0.26. –  goodluck Jan 17 '13 at 9:01
    
@goodluck Is this a formatting issue? You are aware that Sqrt[9 x] will automatically simplify to 3 Sqrt[x], correct? Do you want a formatting rule like this? –  Mr.Wizard Jan 17 '13 at 10:05
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