Hi Monir
I'm sorry, but this time I don't agree with you on any of the points.
Points 1 and 2.
Root(a + bi, n) = Root(r × exp(φi), n) = (r ^ (1/n)) × exp((φ + 2 × π × k) i / n)
where k takes the values 0, 1, ... n-1
The principal value is defined as the one for k=0.
Let's calculate the principal value for your examples:
Example 1: -4+0i
r=4, φ=π, Root(-4 + 0i, 2) = (4^(1/2)) × exp((π + 2 × π × k) i / 2)
Principal value: 2 × exp(πi / 2) = 2i
Example 2: 8+0i
r=8, φ=0, Root(8 + 0i, 3) = (8^(1/3)) × exp((0 + 2 × π × k) i / 3)
Principal value: 2 × exp(0i / 3) = 2
Example 3: -8+0i
r=8, φ=π, Root(-8 + 0i, 3) = (8^(1/3)) × exp((π + 2 × π × k) i / 3)
Principal value: 2 × exp(πi / 3) = 2 × (1/2 + 3^(1/2)/2i) = 1 + 3^(1/2)i
Example 4: -3+4i
r=5, φ=ATAN2(-4/3), Root(-3 + 4i, 3) = (5^(1/3)) × exp((ATAN2(-4/3) + 2 × π × k) i / 3)
Principal value: (5^(1/3)) × exp(ATAN2(-4/3) i / 3) = 1.26495290635775 + 1.15061369838445i
Table with the results calculated here and the results calculated using ImPower():
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PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Input</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Principal</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; 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PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Imaginary</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">ImPower()</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">5</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-4+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">0</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.22514845490862E-16+2i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">6</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">8+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">0</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">7</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-8+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.73205080756888</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1+1.73205080756888i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">8</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-3+4i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.26495290635775</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.15061369838445</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.26495290635775+1.15061369838445i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">9</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; 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BACKGROUND: #9cf" colSpan=10>[book2.xlsb]Sheet2</TD></TR></TBODY></TABLE>
You say that ImPower() doesn't always return the principal value of the root. I don't see where that's happening.
I will answer to the points 3 and 4 later. Let me tell you, however that I disagree completely with them.