Which of the following complex numbers is equivalent to (3-5i)/(8+2i) ? (Note: {:i=sqrt(-1)) A) (3)/(8)-(5i)/(2) B) (3)/(8)+(5i)/(2) C) (7)/(34)-(23 i)/(34) D) (7)/(34)+(23 i)/(34)

Question

Which of the following complex numbers is equivalent to  (3-5i)/(8+2i) ? (Note:  {:i=sqrt(-1)) A)  (3)/(8)-(5i)/(2) B)  (3)/(8)+(5i)/(2) C)  (7)/(34)-(23 i)/(34) D)  (7)/(34)+(23 i)/(34)

Bytelearn · Accepted Answer

Multiply by Conjugate: To simplify the complex fraction (3-5i)/(8+2i), we will multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator. The conjugate of (8+2i) is (8-2i).Expand Numerator: Multiply the numerator and the denominator by the conjugate of the denominator: ((3-5i)(8-2i))/((8+2i)(8-2i))Expand Denominator: Expand the numerator: (3-5i)(8-2i) = 3*8 + 3*(-2i) - 5i*8 - 5i*(-2i) = 24 - 6i - 40i + 10i^2 Since i^2 = -1, we have: = 24 - 46i - 10 = 14 - 46iDivide Numerator by Denominator: Expand the denominator: (8+2i)(8-2i) = 8*8 - 8*2i + 2i*8 - 2i*2i = 64 - 16i + 16i - 4i^2 Since i^2 = -1, we have: = 64 + 4 = 68Simplify Fraction: Now, divide the simplified numerator by the simplified denominator: (14 - 46i) / 68Final Result: Simplify the fraction by dividing both terms by 68: 14/68 - 46i/68 = 7/34 - 23i/34The simplified complex number is 7/34 - 23i/34, which corresponds to option C.

Which of the following complex numbers is equivalent to $(3-5i)/(8+2i)$ ? (Note: $i=\sqrt{-1}$ ) $\newline$ A) $(3)/(8)-(5i)/(2)$ $\newline$ B) $(3)/(8)+(5i)/(2)$ $\newline$ C) $(7)/(34)-(23 i)/(34)$ $\newline$ D) $(7)/(34)+(23 i)/(34)$

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