- Identify the imaginary unit property: \(\sqrt{-k} = i\sqrt{k}\).
- Simplify the first term: \(\sqrt{-16} = 4i\).
- Simplify the second term: \(\sqrt{49} = 7\).
- Combine to get the result: \(\boxed{7+4i}\).
Explanation
- Analysis of the Expression
We need to simplify the expression \(\sqrt{-16} + \sqrt{49}\) and write it in the standard form for complex numbers, \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined as \(i = \sqrt{-1}\).
- Simplifying the First Term
The term \(\sqrt{-16}\) can be rewritten using the property of the imaginary unit: \(\sqrt{-k} = i\sqrt{k}\) for any positive real number \(k\).
Therefore, \(\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\).
- Simplifying the Second Term
The second term \(\sqrt{49}\) is the positive square root of 49. Since \(7 \times 7 = 49\), we have \(\sqrt{49} = 7\).
- Conclusion and Final Result
Combining the two results, we get \(7 + 4i\). This expression is already in the form \(a + bi\), where \(a = 7\) and \(b = 4\). Comparing this result with the given choices, we see that it corresponds to the first option.
Examples
Understanding complex numbers is essential for advanced mathematics and physics. For example, in electrical engineering, complex numbers are used to analyze alternating current (AC) circuits, where resistors, capacitors, and inductors react differently to voltage and current over time. By using complex numbers of the form \(a + bi\), engineers can easily calculate the total impedance of a circuit. Here, \(\sqrt{-16}\) acts as an imaginary component that helps describe the phase shifts in these circuits.