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You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

Edit: To address your question:

{2,1} is a TreeCoordinate of the part of the whole expression that is under the outer square root. The Tree you can visualize by the function

TreeForm[expr]

yielding the following structure

enter image description here

Here the arrow indicates the element {2,1} that wee need. This can be made visible, if you hover the cursor over this element. It is this element that the PowerExpand function is convenient to be applied to.

You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

Edit: To address your question:

{2,1} is a TreeCoordinate of the part of the whole expression that is under the outer square root. The Tree you can visualize by the function

TreeForm[expr]

yielding the following structure

enter image description here

Here the arrow indicates the element {2,1} that wee need. This can be made visible, if you hover the cursor over this element. It is this element that the PowerExpand function is convenient to be applied to.

You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

Edit: To address your question:

{2,1} is a TreeCoordinate of the part of the whole expression that is under the outer square root. The Tree you can visualize by the function

TreeForm[expr]

yielding the following structure

enter image description here

Here the arrow indicates the element {2,1} that wee need. This can be made visible, if you hover the cursor over this element. It is this element that the PowerExpand function is convenient to be applied to.

2 added 546 characters in body
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You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

Edit: To address your question:

{2,1} is a TreeCoordinate of the part of the whole expression that is under the outer square root. The Tree you can visualize by the function

TreeForm[expr]

yielding the following structure

enter image description here

Here the arrow indicates the element {2,1} that wee need. This can be made visible, if you hover the cursor over this element. It is this element that the PowerExpand function is convenient to be applied to.

You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!

Edit: To address your question:

{2,1} is a TreeCoordinate of the part of the whole expression that is under the outer square root. The Tree you can visualize by the function

TreeForm[expr]

yielding the following structure

enter image description here

Here the arrow indicates the element {2,1} that wee need. This can be made visible, if you hover the cursor over this element. It is this element that the PowerExpand function is convenient to be applied to.

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source | link

You have given contradictive assumptions. In Mma the condition that a variable, say, c, is positive (c>0) automatically means that it belongs to Reals. Thus, when you fix c \[Element] Complexes && (a | b | c) > 0 you mislead Mma.

According to your initial expressions the parameters a and b are Reals and positive, while c is Complex, am I right? If yes, try this:

    expr = Simplify[Sqrt[(x /. sol[[2, 1]])^2 + (y /. sol[[2, 2]])^2], 
      Assumptions -> {a, b} > 0];

MapAt[PowerExpand, expr, {2, 1}]

(*  1/2 Sqrt[-((I Sqrt[b^2 - 16 Im[c]^2 - 16 Re[c]^2])/a)]  *)

Have fun!