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I would like to understand why FullSimplify doesn't entirely simplify this expression

FullSimplify[E^(-2*t*g)*Sqrt[E^(4*t*(g + 2*I*l))], {l > 0, g > 0, t > 0}]

simplifies to

Sqrt[E^(4*(g + (2*I)*l)*t)]*E^(-2*g*t)

i.e. it doesn't eliminate the real part of the exponential, while

FullSimplify[ E^(-2*t*g)*Sqrt[E^(4*t*(g + 2*I*l))] == Sqrt[Exp[8 I l t]], {l > 0, g > 0, t > 0}]

yields True.

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    $\begingroup$ This does work ComplexExpand[E^(-2*t*g)*Sqrt[E^(4*t*(g + 2*I*l))]] // FullSimplify[#, {g > 0, t > 0, l > 0}] &. $\endgroup$
    – gwr
    Commented Jun 20, 2015 at 14:45

1 Answer 1

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Use ComplexExpand

expr1 = E^(-2*t*g)*Sqrt[E^(4*t*(g + 2*I*l))];

expr2 = expr1 // ComplexExpand // 
  FullSimplify[#, {l > 0, g > 0, t > 0}] &

Sqrt[E^(8*Ilt)]

expr1 == expr2 // FullSimplify[#, {l > 0, g > 0, t > 0}] &

True

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  • $\begingroup$ +1. I had not seen your answer when I posted my comment. $\endgroup$
    – gwr
    Commented Jun 20, 2015 at 15:36

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