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प्रश्न
Find the square roots of – 6 + 8i
उत्तर
Let `sqrt(- 6 + 8"i")` = a + ib
Squaring on both sides
– 6 + 8i = (a + ib)2
– 6 + 8i = a2 – b2 + 2iab
Equating real and imaginary parts
a2 – b2 = – 6 and 2ab = 8
Now (a2 + b2)2 = (a2 – b2)2 + 4a2b2
= (– 6)2 + (8)2
= 36 + 64 = 100
∴ a + b2 = 10
Solving a2 – b2 = – 6 and a2 + b2 = 10
We get 2a2 = 4, b2 = 8
a2 = 2
b2 = `+- 2sqrt(2)`
a = `+- sqrt(2)`
∴ `sqrt(- 6 + 8"i") = +- sqrt(2) +- "i"2sqrt(2)`
= `+- (sqrt(2) + "i" 2sqrt(2))`
Aliter:
Square root of – 6 + 8i
Let a + ib = – 6 + 8i
a = – 6, b = 8
|z| = `sqrt(6^2 + 8^2)`
= `sqrt(100)`
= 10
`sqrt("a" + "b") = +- [sqrt((sqrt("a"^2 + "b"^2) + "a")/2) + "i" "b"/|"b"| sqrt((sqrt("a"^2 + "b"^2) - "a")/2)]`
= `+- [sqrt((10 - 6)/2) + "i" sqrt((10 + 6)/2)]`
= `+- [sqrt(2) + "i" 2sqrt(2)]`
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