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प्रश्न
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values of a and b: `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
उत्तर
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2) = (4("i"^4)^2 - 3("i"^4)^2*"i" + 3)/(3("i"^4)^2*"i"^3 - 4("i"^4)^2*"i"^2 - 2)`
Since, i2 = – 1, i3 = – i and i4 = 1
∴ `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2) = (4(1)^2 - 3(1)^2*"i" + 3)/(3(1)^2(-"i") - 4(1)^2 (-1) - 2)`
= `(4 - 3"i" + 3)/(-3"i" + 4 - 2)`
= `(7 - 3"i")/(2 - 3"i")`
= `((7 - 3"i")(2 + 3"i"))/((2 - 3"i")(2 + 3"i")`
= `(14 + 21"i" - 6"i" - 9"i"^2)/(4 - 9(-1))`
= `(14 + 15"i" - 9(-1))/(4 - 9(-1)`
= `(23 + 15"i")/13`
∴ `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2) = 23/13 + 15/13 "i"`
∴ a = `23/13 and "b" = 15/13`
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