Simplify with negative radicands in terms of [tex]i: x=\frac{5+\sqrt{-49}}{6}[/tex]
Answer (1)
Rewrite the square root of the negative number using the imaginary unit: − 49 = 7 i .
Substitute this into the expression: x = 6 5 + 7 i .
Separate the fraction into real and imaginary parts: x = 6 5 + 6 7 i .
The simplified expression is 6 5 + 6 7 i .
Explanation
Understanding the Problem We are asked to simplify the expression x = 6 5 + − 49 in t er m so f i . T hi s in v o l v ese x p ress in g t h es q u a reroo t o f an e g a t i v e n u mb er u s in g t h e ima g ina ry u ni t i , w h ere i^2 = -1$.
Simplifying the Square Root First, we rewrite − 49 as 49 ⋅ − 1 . Since 49 = 7 and − 1 = i , we have − 49 = 7 i .
Substituting and Separating Now, substitute this back into the expression for x : x = 6 5 + 7 i .
We separate the fraction into real and imaginary parts: x = 6 5 + 6 7 i .
Rewriting the imaginary part, we get x = 6 5 + 6 7 i .
Final Answer Therefore, the simplified expression is x = 6 5 + 6 7 i .
Comparing this to the given options, we see that it matches the option 6 5 + 6 i 7 if we interpret 6 7 i as 6 i 7 . However, the correct interpretation of 6 7 i is 6 7 i , so we need to check if 6 7 i = 6 i 7 .
To check this, we can multiply both sides by 6 i :
6 i ⋅ 6 7 i = 7 i 2 = 7 ( − 1 ) = − 7 6 i ⋅ 6 i 7 = 7 Since − 7 = 7 , we have 6 7 i = 6 i 7 .
However, the option 6 5 + 6 7 i is equivalent to 6 5 + 6 7 i .
Conclusion Thus, the simplified expression is 6 5 + 6 7 i .
Examples
Complex numbers are used in electrical engineering to analyze AC circuits. The impedance of a circuit element, such as a capacitor or inductor, can be represented as a complex number. By using complex numbers, engineers can easily calculate the voltage and current in AC circuits, which is essential for designing and troubleshooting electronic devices.
