Multiply
a) [tex]$2 i^9 * 3 i^4$[/tex]
b) [tex]$(2 \sqrt{3})(5 \sqrt{12})$[/tex]
c) [tex]$(4+5 i)(2-3 i)$[/tex]
Answer (2)
Simplify i 9 to i and i 4 to 1 , then multiply 2 i ∗ 3 ∗ 1 to get 6 i .
Multiply the coefficients and radicals in ( 2 3 ) ( 5 12 ) to get 10 36 , which simplifies to 10 ∗ 6 = 60 .
Expand ( 4 + 5 i ) ( 2 − 3 i ) using the distributive property, simplify using i 2 = − 1 , and combine real and imaginary parts to get 23 − 2 i .
The final answers are: a) 6 i , b) 60 , c) 23 − 2 i .
Explanation
Problem Analysis We are asked to multiply the given expressions. Let's tackle each one step by step.
Multiplying complex numbers a) We need to multiply 2 i 9 ∗ 3 i 4 . Recall that i = − 1 , so i 2 = − 1 . We can simplify the powers of i as follows:
i 9 = i 8 + 1 = i 8 ∗ i = ( i 2 ) 4 ∗ i = ( − 1 ) 4 ∗ i = 1 ∗ i = i i 4 = ( i 2 ) 2 = ( − 1 ) 2 = 1
Now, we can multiply the expression:
2 i 9 ∗ 3 i 4 = 2 ∗ i ∗ 3 ∗ 1 = 6 i
Multiplying radicals b) We need to multiply ( 2 3 ) ( 5 12 ) . We can multiply the coefficients and the square roots separately:
( 2 3 ) ( 5 12 ) = 2 ∗ 5 ∗ 3 ∗ 12 = 10 ∗ 3 ∗ 12 = 10 ∗ 36 = 10 ∗ 6 = 60
Multiplying complex numbers c) We need to multiply ( 4 + 5 i ) ( 2 − 3 i ) . We can use the distributive property (FOIL method) to expand the product:
( 4 + 5 i ) ( 2 − 3 i ) = 4 ∗ 2 + 4 ∗ ( − 3 i ) + 5 i ∗ 2 + 5 i ∗ ( − 3 i ) = 8 − 12 i + 10 i − 15 i 2
Since i 2 = − 1 , we have:
8 − 12 i + 10 i − 15 i 2 = 8 − 12 i + 10 i − 15 ( − 1 ) = 8 − 12 i + 10 i + 15 = ( 8 + 15 ) + ( − 12 + 10 ) i = 23 − 2 i
Final Answer Therefore, the results are: a) 6 i b) 60 c) 23 − 2 i
Examples
Complex numbers and radical multiplication are used in electrical engineering to analyze AC circuits, in quantum mechanics to describe wave functions, and in signal processing. For example, when analyzing an AC circuit, complex numbers help represent the impedance, which combines resistance and reactance. Multiplying these complex impedances helps determine the overall behavior of the circuit. Similarly, multiplying radicals is essential in calculating areas and volumes in geometry and physics, such as determining the volume of a sphere or the area of a triangle.
The results of the multiplications are: a) 6 i , b) 60 , c) 23 − 2 i . Each calculation involved simplifying the expressions using properties of imaginary numbers and radicals. The answers illustrate how they can be approached step by step for clarity in complex arithmetic.
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