What is the sum of [tex]$\sqrt{-2}$[/tex] and [tex]$\sqrt{-18}$[/tex]?
Answer (2)
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 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
Imaginary numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing alternating current (AC) circuits, imaginary numbers help represent the phase difference between voltage and current. By using complex numbers, engineers can simplify calculations and design more efficient circuits. This is just one example of how imaginary numbers, which start with seemingly simple expressions like − 1 , play a crucial role in real-world applications.
The sum of − 2 and − 18 is 4 2 i after simplifying each term using the imaginary unit i . This involves expressing each square root in terms of i and then combining the results. Thus, the answer is 4 2 i .
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