Simplify $-\sqrt{-48}$.
A) $4 \sqrt{3} i$
B) $-4 \sqrt{3}$
C) $-4 \sqrt{3} i$
D) $4 \sqrt{3}$
Answer (2)
Rewrite the expression: − − 48 = − ( − 1 ) ( 48 ) .
Separate the square root: − − 1 ⋅ 48 .
Replace − 1 with i : − i ⋅ 48 .
Simplify 48 : − i ⋅ 4 3 = − 4 3 i . The final answer is − 4 3 i .
Explanation
Understanding the Problem We are asked to simplify the expression − − 48 . This involves dealing with the square root of a negative number, which means we'll be working with imaginary numbers.
Rewriting the Expression First, we can rewrite the expression as follows: − − 48 = − ( − 1 ) ( 48 ) This separates the negative sign from the number under the square root.
Separating the Square Root Next, we can separate the square root into two parts: − − 1 ⋅ 48 We know that − 1 is defined as the imaginary unit i .
Simplifying the Square Root of 48 Now, let's simplify 48 . We look for the largest perfect square that divides 48. The largest perfect square factor of 48 is 16, since 48 = 16 ⋅ 3 . Therefore, we can write: 48 = 16 ⋅ 3 = 16 ⋅ 3 = 4 3
Final Simplification Now, we substitute these simplified parts back into the expression: − i ⋅ 4 3 = − 4 3 i So, the simplified form of − − 48 is − 4 3 i .
Final Answer Therefore, the simplified expression is − 4 3 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, if you're designing a circuit and need to calculate the impedance (resistance to AC), you'll often use complex numbers (which include imaginary parts) to account for the effects of inductors and capacitors. This ensures the circuit functions correctly and efficiently.
The expression − − 48 simplifies to − 4 3 i by recognizing that − 1 is the imaginary unit i and simplifying 48 . Thus, the correct choice is option C: − 4 3 i .
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