Find the value of: $\frac{2-j}{3+j}+\frac{2+j}{3-j}$
Answer (1)
Find a common denominator and combine the fractions: 3 + j 2 − j + 3 − j 2 + j = ( 3 + j ) ( 3 − j ) ( 2 − j ) ( 3 − j ) + ( 2 + j ) ( 3 + j ) .
Expand the numerator: ( 2 − j ) ( 3 − j ) + ( 2 + j ) ( 3 + j ) = 5 − 5 j + 5 + 5 j = 10 .
Expand the denominator: ( 3 + j ) ( 3 − j ) = 9 − j 2 = 9 − ( − 1 ) = 10 .
Simplify the expression: 10 10 = 1 . The final answer is 1 .
Explanation
Problem Analysis We are asked to find the value of the expression 3 + j 2 − j + 3 − j 2 + j , where j is the imaginary unit, i.e., j = − 1 .
Finding Common Denominator To solve this, we will first find a common denominator and add the two fractions. The common denominator is ( 3 + j ) ( 3 − j ) . Thus, we have 3 + j 2 − j + 3 − j 2 + j = ( 3 + j ) ( 3 − j ) ( 2 − j ) ( 3 − j ) + ( 2 + j ) ( 3 + j ) .
Expanding Numerator and Denominator Next, we expand the numerator and the denominator.
Numerator: ( 2 − j ) ( 3 − j ) = 2 ( 3 ) + 2 ( − j ) − j ( 3 ) + ( − j ) ( − j ) = 6 − 2 j − 3 j + j 2 = 6 − 5 j − 1 = 5 − 5 j ( 2 + j ) ( 3 + j ) = 2 ( 3 ) + 2 ( j ) + j ( 3 ) + j ( j ) = 6 + 2 j + 3 j + j 2 = 6 + 5 j − 1 = 5 + 5 j So, ( 2 − j ) ( 3 − j ) + ( 2 + j ) ( 3 + j ) = ( 5 − 5 j ) + ( 5 + 5 j ) = 10 .
Denominator: ( 3 + j ) ( 3 − j ) = 3 ( 3 ) + 3 ( − j ) + j ( 3 ) + j ( − j ) = 9 − 3 j + 3 j − j 2 = 9 − ( − 1 ) = 10.
Substituting and Simplifying Now we substitute the expanded expressions back into the equation: ( 3 + j ) ( 3 − j ) ( 2 − j ) ( 3 − j ) + ( 2 + j ) ( 3 + j ) = 10 10 = 1.
Final Answer Therefore, the value of the expression is 1.
Examples
Complex numbers are used in electrical engineering to represent alternating currents. The expression in this problem could represent the impedance of a circuit, and finding its value helps engineers understand the behavior of the circuit.
