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Express (2+j3)/(j(4-j5)) + 2/j in the form a+jb.

Mathematics

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: Express 2+j3j(4−j5)+2j\frac{2+j3}{j(4-j5)} + \frac{2}{j} in the form a+jba+jb.

  1. Simplify the denominator of the first term and the second term.

    For the first term's denominator:

    j(4−j5)=j4−j25=j4−(−1)5=5+j4j(4-j5) = j4 - j^2 5 = j4 - (-1)5 = 5+j4

    For the second term:

    2j=2j×−j−j=−j2−j2=−j2−(−1)=−j21=−j2\frac{2}{j} = \frac{2}{j} \times \frac{-j}{-j} = \frac{-j2}{-j^2} = \frac{-j2}{-(-1)} = \frac{-j2}{1} = -j2
  2. Simplify the first term, 2+j35+j4\frac{2+j3}{5+j4}, by multiplying the numerator and denominator by the conjugate of the denominator, which is 5−j45-j4.

    2+j35+j4=2+j35+j4×5−j45−j4\frac{2+j3}{5+j4} = \frac{2+j3}{5+j4} \times \frac{5-j4}{5-j4}

    Numerator:

    (2+j3)(5−j4)=2(5)+2(−j4)+j3(5)+j3(−j4)(2+j3)(5-j4) = 2(5) + 2(-j4) + j3(5) + j3(-j4) =10−j8+j15−j212= 10 - j8 + j15 - j^2 12 =10+j7−(−1)12= 10 + j7 - (-1)12 =10+j7+12=22+j7= 10 + j7 + 12 = 22 + j7

    Denominator:

    (5+j4)(5−j4)=52−(j4)2(5+j4)(5-j4) = 5^2 - (j4)^2 =25−j216= 25 - j^2 16 =25−(−1)16= 25 - (-1)16 =25+16=41= 25 + 16 = 41

    So the first term simplifies to 22+j741=2241+j741\frac{22+j7}{41} = \frac{22}{41} + j\frac{7}{41}.

  3. Add the simplified first term and the simplified second term.

    (2241+j741)+(−j2)\left(\frac{22}{41} + j\frac{7}{41}\right) + (-j2) =2241+j741−j8241= \frac{22}{41} + j\frac{7}{41} - j\frac{82}{41} =2241+j(7−8241)= \frac{22}{41} + j\left(\frac{7-82}{41}\right) =2241−j7541= \frac{22}{41} - j\frac{75}{41}

    The expression in the form a+jba+jb is 2241−j7541\boxed{\frac{\text{22}}{\text{41}} - j\frac{\text{75}}{\text{41}}}.

    : If z=12+j3+11−j2z = \frac{1}{2+j3} + \frac{1}{1-j2}, express zz in the form a+jba+jb.

  4. Simplify the first term, 12+j3\frac{1}{2+j3}, by multiplying the numerator and denominator by its conjugate, 2−j32-j3.

    12+j3=12+j3×2−j32−j3\frac{1}{2+j3} = \frac{1}{2+j3} \times \frac{2-j3}{2-j3} =2−j322−(j3)2=2−j34−j29=2−j34−(−1)9=2−j34+9=2−j313=213−j313= \frac{2-j3}{2^2 - (j3)^2} = \frac{2-j3}{4 - j^2 9} = \frac{2-j3}{4 - (-1)9} = \frac{2-j3}{4+9} = \frac{2-j3}{13} = \frac{2}{13} - j\frac{3}{13}
  5. Simplify the second term, 11−j2\frac{1}{1-j2}, by multiplying the numerator and denominator by its conjugate, 1+j21+j2.

    11−j2=11−j2×1+j21+j2\frac{1}{1-j2} = \frac{1}{1-j2} \times \frac{1+j2}{1+j2} =1+j212−(j2)2=1+j21−j24=1+j21−(−1)4=1+j21+4=1+j25=15+j25= \frac{1+j2}{1^2 - (j2)^2} = \frac{1+j2}{1 - j^2 4} = \frac{1+j2}{1 - (-1)4} = \frac{1+j2}{1+4} = \frac{1+j2}{5} = \frac{1}{5} + j\frac{2}{5}
  6. Add the simplified first and second terms.

    z=(213−j313)+(15+j25)z = \left(\frac{2}{13} - j\frac{3}{13}\right) + \left(\frac{1}{5} + j\frac{2}{5}\right)

    Combine the real parts and the imaginary parts:

    z=(213+15)+j(−313+25)z = \left(\frac{2}{13} + \frac{1}{5}\right) + j\left(-\frac{3}{13} + \frac{2}{5}\right)

    Find a common denominator for the real parts (13 and 5, LCM is 65):

    213+15=2×513×5+1×135×13=1065+1365=2365\frac{2}{13} + \frac{1}{5} = \frac{2 \times 5}{13 \times 5} + \frac{1 \times 13}{5 \times 13} = \frac{10}{65} + \frac{13}{65} = \frac{23}{65}

    Find a common denominator for the imaginary parts (13 and 5, LCM is 65):

    −313+25=−3×513×5+2×135×13=−1565+2665=1165-\frac{3}{13} + \frac{2}{5} = \frac{-3 \times 5}{13 \times 5} + \frac{2 \times 13}{5 \times 13} = \frac{-15}{65} + \frac{26}{65} = \frac{11}{65}

    So, z=2365+j1165z = \frac{23}{65} + j\frac{11}{65}.
    The expression for zz in the form a+jba+jb is 2365+j1165\boxed{\frac{\text{23}}{\text{65}} + j\frac{\text{11}}{\text{65}}}.

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