Simplify with negative radicands in terms of $i$: [tex]x=\frac{5+\sqrt{-49}}{6}[/tex]
Answer (1)
Simplify the square root of the negative number: − 49 = 7 i .
Substitute the simplified square root back into the expression: x = 6 5 + 7 i .
Separate the 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 given the expression x = 6 5 + − 49 and asked to simplify it in terms of i .
Simplifying the Radicand First, we need to simplify the square root of a negative number. Recall that − 1 = i . Therefore, we can rewrite − 49 as 49 × − 1 = 49 × − 1 .
Evaluating the Square Root Since 49 = 7 and − 1 = i , we have − 49 = 7 i .
Substituting Back into the Expression Now, substitute this back into the original expression: x = 6 5 + 7 i .
Separating Real and Imaginary Parts Finally, we separate the real and imaginary parts by dividing both terms in the numerator by 6: x = 6 5 + 6 7 i = 6 5 + 6 7 i .
Final Answer Thus, the simplified expression is 6 5 + 6 7 i .
Examples
Complex numbers, like the one we just simplified, are used in electrical engineering to analyze AC circuits. The impedance of a circuit, which is the opposition to the flow of current, is often expressed as a complex number. Simplifying expressions with complex numbers helps engineers understand and design these circuits effectively. For example, the expression could represent the voltage across a component in an AC circuit, and simplifying it allows engineers to calculate the current flowing through that component.
