Add:
a) $(3+2 i)+(5+6 i)$
b) $2 \sqrt{5}+7 \sqrt{5}$
c) $\sqrt{-27}+\sqrt{-12}$
Answer (2)
Add complex numbers by summing real and imaginary parts: ( 3 + 2 i ) + ( 5 + 6 i ) = 8 + 8 i .
Add radical expressions with like radicands by summing coefficients: 2 5 + 7 5 = 9 5 .
Simplify radicals with negative radicands using i = − 1 and then add: − 27 + − 12 = 5 i 3 .
The final answers are: a) 8 + 8 i , b) 9 5 , c) 5 i 3 .
Explanation
Problem Analysis We are asked to perform addition on three separate problems: a) Add two complex numbers. b) Add two radical expressions with the same radicand. c) Add two radical expressions with negative radicands.
Adding Complex Numbers a) To add complex numbers, we add the real parts together and the imaginary parts together: ( 3 + 2 i ) + ( 5 + 6 i ) = ( 3 + 5 ) + ( 2 + 6 ) i = 8 + 8 i
Adding Radical Expressions b) To add radical expressions with the same radicand, we add their coefficients: 2 5 + 7 5 = ( 2 + 7 ) 5 = 9 5
Adding Radical Expressions with Negative Radicands c) To add radical expressions with negative radicands, we first simplify each radical by expressing the negative radicand in terms of i = − 1 :
− 27 = 27 ⋅ − 1 = 9 ⋅ 3 ⋅ i = 3 3 i − 12 = 12 ⋅ − 1 = 4 ⋅ 3 ⋅ i = 2 3 i Now, we add the simplified expressions: 3 3 i + 2 3 i = ( 3 + 2 ) 3 i = 5 3 i
Final Answer The results of the additions are: a) 8 + 8 i b) 9 5 c) 5 i 3
Examples
Complex numbers are used in electrical engineering to represent alternating currents. Radical expressions are used in various fields such as physics and engineering to calculate distances, areas, and volumes. Understanding how to add these types of numbers is crucial for solving problems in these fields. For example, when analyzing AC circuits, engineers often need to add complex impedances, which are represented as complex numbers. Similarly, in geometry, adding radical expressions might be necessary when calculating the perimeter of a figure with sides of irrational lengths.
We have added three expressions: for part a, the result is 8 + 8 i ; for part b, the result is 9 5 ; and for part c, the result is 5 3 i .
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