Verified Answer :

  • Recognize that the standard form of a complex number is \(a + bi\), where \(i = \sqrt{-1}\).
  • Simplify the square root of the negative number using the property \(\sqrt{-x} = i\sqrt{x}\), yielding \(\sqrt{-9} = 3i\).
  • Combine the real part with the simplified imaginary part to reach the final form \(-1 + 3i\).
  • Final Answer: \(\boxed{-1+3i}\)

Explanation

  1. Problem Analysis

We are tasked with expressing the expression \(-1 + \sqrt{-9}\) in the standard form of a complex number, which is \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined as \(\sqrt{-1}\). Our objective is to simplify the term \(\sqrt{-9}\) and combine it with the real part \(-1\).

  1. Simplifying the Imaginary Term

To simplify the square root of a negative number, we use the property \(\sqrt{-x} = i\sqrt{x}\) for any positive real number \(x\). Applying this to our expression, we have \(\sqrt{-9} = i\sqrt{9}\). Since \(\sqrt{9} = 3\), we can simplify this further:
\[\sqrt{-9} = i \times 3 = 3i\]

  1. Formulating the Standard Complex Form

Now that we have simplified the imaginary part, we substitute it back into the original expression:
\[-1 + \sqrt{-9} = -1 + 3i\]
This expression is now in the required standard complex form \(a + bi\), where \(a = -1\) and \(b = 3\). Thus, the final form is \(-1 + 3i\).

Examples

Complex numbers are used extensively in fields like electrical engineering and physics. For example, when analyzing alternating current (AC) circuits, voltages and currents are represented as complex numbers to easily handle phase differences between them. The imaginary component represents the reactive power, while the real component corresponds to the real power dissipated, allowing engineers to simplify complex circuit calculations into standard algebraic operations.

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