Simplify with negative radicands in terms of $i$:
$x=\frac{5+\sqrt{-49}}{6}$
Answer (2)
Rewrite − 49 as 7 i .
Substitute 7 i 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 asked to simplify the expression x = 6 5 + − 49 in terms of i . Recall that i is the imaginary unit, defined as i = − 1 .
Simplifying the Square Root First, we need to simplify the square root of the negative number. We can rewrite − 49 as 49 ⋅ − 1 . Using the property of square roots, we have 49 ⋅ − 1 = 49 ⋅ − 1 .
Substituting Known Values Since 49 = 7 and − 1 = i , we can substitute these values to get − 49 = 7 i .
Substituting into the Original Expression Now, substitute 7 i for − 49 in the expression for x : x = 6 5 + 7 i .
Final Simplification Finally, we can rewrite the expression as x = 6 5 + 6 7 i , which is the same as x = 6 5 + 6 7 i . So the simplified expression is 6 5 + 6 7 i .
Examples
Complex numbers, involving the imaginary unit i , are not just abstract mathematical concepts; they have practical applications in various fields. For example, in electrical engineering, complex numbers are used to analyze alternating current (AC) circuits. The impedance, which is the opposition to the flow of current in an AC circuit, is represented as a complex number. The real part of the impedance represents the resistance, while the imaginary part represents the reactance. By using complex numbers, engineers can easily calculate the voltage and current in AC circuits. Another application is in quantum mechanics, where the wave function of a particle is a complex-valued function.
To simplify x = 6 5 + − 49 , we replace − 49 with 7 i , leading to x = 6 5 + 7 i . The final simplified expression is 6 5 + 6 7 i . Therefore, the answer is 6 5 + 6 7 i .
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