Add as indicated.
$(5+i \sqrt{6})+(5+7 i \sqrt{6}) = $
$(5+i \sqrt{6}) *(5 + 7 i \sqrt{6}) = $
$\square$
(Simplify your answer. Type an exact answer, using radicals as needed.)
Answer (1)
10 + 8 i 6 , − 210 + 175 i 6
Explanation
Problem Analysis We are asked to add and multiply complex numbers. Let's start by analyzing the given expressions.
The first expression is a sum of two complex numbers: ( 5 + i 6 ) + ( 5 + 7 i 6 ) .
The second expression is a product of a complex number and a real number: ( 5 + i 6 ) ∗ ( 5 ∗ 7 i 6 ) .
Addition of Complex Numbers To add the complex numbers, we add the real parts and the imaginary parts separately: ( 5 + i 6 ) + ( 5 + 7 i 6 ) = ( 5 + 5 ) + ( i 6 + 7 i 6 ) = 10 + 8 i 6 So the first expression simplifies to 10 + 8 i 6 .
Multiplication of Complex Numbers For the second expression, we first simplify the term in the second parenthesis: 5 ∗ 7 i 6 = 35 i 6 Now, we multiply the complex number ( 5 + i 6 ) by 35 i 6 :
( 5 + i 6 ) ∗ ( 35 i 6 ) = 5 ∗ ( 35 i 6 ) + i 6 ∗ ( 35 i 6 ) = 175 i 6 + 35 ∗ i 2 ∗ ( 6 ) 2 Since i 2 = − 1 and ( 6 ) 2 = 6 , we have: 175 i 6 + 35 ∗ ( − 1 ) ∗ 6 = 175 i 6 − 210 = − 210 + 175 i 6 So the second expression simplifies to − 210 + 175 i 6 .
Final Answer Therefore, the simplified expressions are:
Addition: 10 + 8 i 6 Multiplication: − 210 + 175 i 6
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The voltage and current in an AC circuit can be represented as complex numbers, where the real part represents the resistive component and the imaginary part represents the reactive component. By using complex numbers, engineers can easily analyze and design AC circuits. For example, the impedance of a circuit, which is the opposition to the flow of current, can be calculated using complex numbers.
