Simplify
a) [tex]i^{21}[/tex]
b) [tex]\sqrt{-49}[/tex]
c) [tex]\frac{-12 \pm \sqrt{-18}}{3}[/tex]
Answer (2)
Simplify i 21 to i .
Simplify − 49 to 7 i .
Simplify 3 − 12 ± − 18 to − 4 ± i 2 .
The simplified expressions are: i , 7 i , and − 4 ± i 2 , respectively. i , 7 i , − 4 ± i 2
Explanation
Problem Analysis We are asked to simplify expressions involving complex numbers. Let's tackle each part step by step.
Simplifying i^21 a) Simplify i 21 . Recall that i = − 1 , i 2 = − 1 , i 3 = − i , and i 4 = 1 . We can write i 21 as i 4 ⋅ 5 + 1 = ( i 4 ) 5 ⋅ i 1 = ( 1 ) 5 ⋅ i = i .
Simplifying sqrt(-49) b) Simplify − 49 . We know that − a = i a . Therefore, − 49 = 49 ⋅ − 1 = 49 ⋅ − 1 = 7 i .
Simplifying the Fraction c) Simplify 3 − 12 ± − 18 . First, let's simplify the square root: − 18 = 18 ⋅ − 1 = 9 ⋅ 2 ⋅ i = 3 i 2 . Now, substitute this back into the expression: 3 − 12 ± 3 i 2 . Finally, divide both terms in the numerator by 3: 3 − 12 ± 3 3 i 2 = − 4 ± i 2 .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. They help in representing impedance, which is the opposition to the flow of current in an AC circuit. By using complex numbers, engineers can simplify calculations and understand the behavior of these circuits more effectively. For example, the voltage and current in an AC circuit can be represented as complex numbers, and their relationship can be analyzed using complex algebra.
The simplifications yield: i 21 = i , sqrt ( − 49 ) = 7 i , and 3 − 12 pm sqrt ( − 18 ) = − 4 pm i sqrt ( 2 ) .
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