I am going to build an inverted V for 80 and 40 meters. I will be
running
about 700 watts into it. What gauge wire should I use and should it
be
stranded or solid core for best performance and power handling?

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Use whatever wire gauge will support its own weight. For 1 Kwatt,
anywhere from 18 awg to 12 awg will do fine.

The RF loss in the wire is very slightly greater with the thinner
gauges, but not enough to be noticeable. In terms of signal strength
it is in terms of hundredths of a decibel.

I prefer solid core, ordinary magnet wire. The enamel covering is a
protection against attack from atmospheric polution. Wires last for
years and years in the British average sort of climate.

The so-called "stretch" in annealed, enamelled, copper wires is not
worth the trouble of mentioning. Just give the wire a good tug during
erection, you will feel it give an inch or so, and unless there's a
double hurricane it will remain the same length for ever afterwards.
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Reg, G4FGQ