Simplify the expression. Write your answer as a complex number.
[tex]2 \sqrt{25}+\sqrt{-16}[/tex]
Answer (2)
The simplified expression of 2 25 + − 16 is 10 + 4 i . First, we evaluate 25 as 5 and − 16 as 4 i . Combining these results leads to the final answer of 10 + 4 i .
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Evaluate the square root of 25: 25 = 5 .
Evaluate the square root of -16: − 16 = 4 i .
Substitute the values back into the expression: 2 ( 5 ) + 4 i .
Simplify the expression to get the final answer: 10 + 4 i .
Explanation
Understanding the problem We are asked to simplify the expression 2 25 + − 16 and write the answer as a complex number.
Evaluating the square root of 25 First, we need to evaluate the square roots. We know that 25 = 5 .
Evaluating the square root of -16 Next, we evaluate − 16 . Since we are looking for a complex number, we can rewrite this as − 16 = 16 ⋅ − 1 = 4 i .
Substituting the values Now, we substitute these values back into the original expression: 2 25 + − 16 = 2 ( 5 ) + 4 i .
Simplifying the expression Finally, we simplify the expression: 2 ( 5 ) + 4 i = 10 + 4 i . So, the simplified expression is 10 + 4 i .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The impedance of a circuit, which is the opposition to the flow of current, is represented as a complex number. For example, if a circuit has a resistance of 10 ohms and a reactance of 4 ohms, the impedance can be represented as 10 + 4i. This allows engineers to easily calculate the current and voltage in the circuit using complex number arithmetic.
