Simplify the expression and choose the correct answer:
[tex]$3 \sqrt{-49}+2 \sqrt{-36}$[/tex]
A. 33
B. -33
C. [tex]$33 i$[/tex]
D. [tex]$-33 i$[/tex]
Answer (2)
The simplified expression for 3 − 49 + 2 − 36 is 33 i . Thus, the correct answer is option C. This result utilizes the imaginary unit to express the square roots of negative numbers.
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Rewrite the expression using the imaginary unit i : 3 − 49 + 2 − 36 = 3 49 ⋅ − 1 + 2 36 ⋅ − 1 .
Simplify the square roots: 3 ⋅ 7 ⋅ i + 2 ⋅ 6 ⋅ i = 21 i + 12 i .
Combine the terms: 21 i + 12 i = ( 21 + 12 ) i = 33 i .
The simplified expression is 33 i .
Explanation
Understanding the Problem We are asked to simplify the expression 3 − 49 + 2 − 36 and choose the correct answer from the provided options.
Using Imaginary Unit To simplify the expression, we need to remember that the square root of a negative number can be expressed using the imaginary unit i , where i = − 1 .
Rewriting the Expression Now, let's rewrite the expression using the imaginary unit i :
3 − 49 + 2 − 36 = 3 49 ⋅ − 1 + 2 36 ⋅ − 1 = 3 49 ⋅ − 1 + 2 36 ⋅ − 1 = 3 ⋅ 7 ⋅ i + 2 ⋅ 6 ⋅ i
Simplifying Simplify the expression:
3 ⋅ 7 ⋅ i + 2 ⋅ 6 ⋅ i = 21 i + 12 i
Combining Terms Combine the terms:
21 i + 12 i = ( 21 + 12 ) i = 33 i
Final Answer Therefore, the simplified expression is 33 i .
Examples
Imaginary numbers might seem abstract, but they're incredibly useful in electrical engineering. When analyzing alternating current (AC) circuits, imaginary numbers help represent the phase difference between voltage and current. For example, the impedance of a circuit, which is the AC equivalent of resistance, is often expressed as a complex number with a real part (resistance) and an imaginary part (reactance). This allows engineers to easily calculate the behavior of circuits with capacitors and inductors.
