Extracting the Square Root of a Voltage
An easy approach would be to use an analog multiplier (MC1495 was an early one, Analog Devices AD633 or Burr-Brown(oops, Texas Instruments!) MPY534 are better newer ones) as a squaring circuit, in the feedback loop of an op-amp. 

To use a multiplier to square a voltage simply connect that voltage to both inputs. Connect your input voltage to opamp's non-inverting input, the opamp output to the multiply inputs and the multiply output to opamp's inverting input.

If \$ V_{out}^2 = V_{in} \$ then \$ V_{out} = \sqrt{V_{in}} \$.







simulate this circuit &ndash; Schematic created using CircuitLab

Details such as DC biasing left as an exercise...

(Side note : the analog multipliers rely heavily on "matched pairs" of transistors; it is relatively easy to match 2 transistors if you make both at once in the same area on the same chip!)