- Simplify the real part: \(\sqrt{64} = 8\).
- Simplify the imaginary part: \(\sqrt{-289} = 17i\).
- Combine the results: \(8 - 17i\).
- Final answer: \(\boxed{8 - 17i}\).
Explanation
- Problem Analysis
We need to express the given expression \(\sqrt{64} - \sqrt{-289}\) in the standard complex number form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined as \(\sqrt{-1}\). The expression consists of a real component \(\sqrt{64}\) and a component involving the square root of a negative number, \(\sqrt{-289}\).
- Simplification of Terms
First, we simplify the real part \(\sqrt{64}\). Since \(8^2 = 64\), we have:
\[\sqrt{64} = 8\]
Next, we simplify the imaginary part \(\sqrt{-289}\). Using the property \(\sqrt{-k} = i\sqrt{k}\) for a positive real number \(k\), we can write:
\[\sqrt{-289} = \sqrt{289} \cdot \sqrt{-1}\]
Knowing that \(17^2 = 289\), it follows that \(\sqrt{289} = 17\). Substituting this, we get:
\[\sqrt{-289} = 17 \cdot i = 17i\]
- Final Combination
Now, we substitute these simplified values back into the original expression:
\[\sqrt{64} - \sqrt{-289} = 8 - 17i\]
This expression is now in the required form \(a + bi\), where \(a = 8\) and \(b = -17\). Comparing this to the given options, we find it corresponds to the third option.
Examples
Complex numbers are used extensively in fields like electrical engineering to represent alternating current (AC) circuit analysis. For instance, impedance is often expressed as a complex number \(Z = R + jX\), where \(R\) is the resistance and \(X\) is the reactance. By performing arithmetic with complex numbers, engineers can calculate the total impedance of circuits involving resistors, capacitors, and inductors, allowing for the design and analysis of complex electronic systems.