Simplify with negative radicands in terms of $i$:
$x=\frac{5+\sqrt{-49}}{6}$
A. $-1 / 3 i$
B. $2 i$
C. $5 / 6+7 i / 6$
D. $5 / 6-7 i / 6$
Answer (1)
∙ Rewrite the square root of the negative number using i : − 49 = 7 i .
∙ Substitute back into the original 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 we want 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 . 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 Separate the real and imaginary parts: 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 alternating current (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 imaginary units helps engineers design and analyze these circuits effectively, ensuring devices function as intended and are safe to use. This is crucial in designing everything from power grids to the circuits in our smartphones.
