What is the sum of [tex]$\sqrt{-2}$[/tex] and [tex]$\sqrt{-18}$[/tex] ?
A. [tex]$4 \sqrt{2}$[/tex]
B. [tex]$4 i \sqrt{2}$[/tex]
C. [tex]$5 \sqrt{2}$[/tex]
D. [tex]$5 i \sqrt{2}$[/tex]
Answer (1)
Express − 2 as 2 i .
Express − 18 as 3 2 i .
Add the two expressions: 2 i + 3 2 i = 4 2 i .
The sum of − 2 and − 18 is 4 i 2 .
Explanation
Understanding the problem We are asked to find the sum of − 2 and − 18 . Since we are dealing with square roots of negative numbers, we will need to use the imaginary unit i , where i = − 1 .
Simplifying − 2 First, let's simplify − 2 . We can rewrite this as 2 × − 1 = 2 × − 1 = 2 i .
Simplifying − 18 Next, let's simplify − 18 . We can rewrite this as 18 × − 1 = 18 × − 1 . Since 18 = 9 × 2 , we have 18 = 9 × 2 = 9 × 2 = 3 2 . Therefore, − 18 = 3 2 i .
Adding the expressions Now, we add the two simplified expressions: − 2 + − 18 = 2 i + 3 2 i . We can factor out 2 i to get ( 1 + 3 ) 2 i = 4 2 i .
Final Answer Therefore, the sum of − 2 and − 18 is 4 2 i .
Examples
Complex numbers, which include imaginary numbers, are used in electrical engineering to analyze alternating current circuits. The impedance of a circuit, which is the opposition to the flow of current, is a complex number. Imaginary numbers are also used in quantum mechanics, a branch of physics that deals with the behavior of matter at the atomic and subatomic levels. They are used to describe the wave function of a particle, which is a mathematical function that describes the probability of finding the particle at a certain location.
