Simplify.
$\begin{array}{c}
(\sqrt{-5}+2 \sqrt{20})(-\sqrt{-5}) \\{[?]+\square i}\end{array}$
Answer (1)
Simplify − 5 as i 5 and 2 20 as 4 5 .
Substitute into the expression: ( i 5 + 4 5 ) ( − − 5 ) = ( i 5 + 4 5 ) ( − i 5 ) .
Distribute and simplify: − i 2 ( 5 ) − 4 i ( 5 ) = 5 − 20 i .
The simplified expression is 5 − 20 i .
Explanation
Understanding the Problem We are asked to simplify the expression ( − 5 + 2 20 ) ( − − 5 ) and express it in the form a + bi , where a and b are real numbers.
Simplifying the Terms First, let's simplify the terms inside the parenthesis. We know that − 5 = 5 ⋅ − 1 = i 5 . Also, 2 20 = 2 4 ⋅ 5 = 2 ⋅ 2 5 = 4 5 .
Substituting the Simplified Terms Now, substitute these simplified expressions back into the original expression: ( i 5 + 4 5 ) ( − i 5 )
Distributing Next, distribute the term − i 5 across the terms inside the parenthesis: ( i 5 ) ( − i 5 ) + ( 4 5 ) ( − i 5 )
Simplifying Each Term Now, simplify each term: For the first term: ( i 5 ) ( − i 5 ) = − i 2 ( 5 ) 2 = − ( − 1 ) ( 5 ) = 5 For the second term: ( 4 5 ) ( − i 5 ) = − 4 i ( 5 ) 2 = − 4 i ( 5 ) = − 20 i
Combining Terms Combine the simplified terms: 5 − 20 i
Final Answer The simplified expression is in the form a + bi , where a = 5 and b = − 20 .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The voltage and current in these circuits can be represented as complex numbers, and the impedance of circuit components (resistors, capacitors, and inductors) can also be represented as complex numbers. By using complex numbers, engineers can easily calculate the behavior of AC circuits.
