Question

name: ____ period:

1. find the value of \\(\sqrt{- 100}\\)

a. -10i
b. 10i
c. 10
d. 100i

1. what is the most simplified version of:

\\(3 + \sqrt{- 4}\\)

a. 5 + i
b. 3 + 2i
c. 5i
d. 3 + i\\(\sqrt{4}\\)

1. what is the value of \\(i^{19}\\)

a. i
b. -1
c. 1
d. -i

Explanation:

Question 1

Step1: Recall the imaginary unit definition

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-100} \) as \( \sqrt{100 \times (-1)} \).

Step2: Use square - root property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \), and here we extend it to the complex number system), we have \( \sqrt{100\times(-1)}=\sqrt{100}\times\sqrt{-1} \).
Since \( \sqrt{100} = 10 \) and \( \sqrt{-1}=i \), then \( \sqrt{-100}=10i \).

Step1: Simplify the square - root of a negative number

First, simplify \( \sqrt{-4} \). We can write \( \sqrt{-4}=\sqrt{4\times(-1)} \).
Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (in the complex number system), we get \( \sqrt{4\times(-1)}=\sqrt{4}\times\sqrt{-1} \).
Since \( \sqrt{4} = 2 \) and \( \sqrt{-1}=i \), then \( \sqrt{-4}=2i \).

Step2: Combine with the real part

Now, we have the expression \( 3+\sqrt{-4} \). Substituting \( \sqrt{-4} = 2i \) into the expression, we get \( 3 + 2i \).

Step1: Recall the pattern of powers of \( i \)

The powers of \( i \) have a cyclic pattern:
\( i^1=i \)
\( i^2=-1 \)
\( i^3=i^2\times i=-1\times i=-i \)
\( i^4=(i^2)^2=(-1)^2 = 1 \)
And then the pattern repeats every 4 powers: \( i^{4n + r}=i^r \), where \( n \) is an integer and \( r\in\{0,1,2,3\} \).

Step2: Divide the exponent by 4

We want to find the value of \( i^{19} \). Divide 19 by 4: \( 19 = 4\times4+3 \).
So, \( i^{19}=i^{4\times4 + 3}=(i^4)^4\times i^3 \).
Since \( i^4 = 1 \), then \( (i^4)^4=1^4 = 1 \), and \( i^3=-i \). So \( i^{19}=1\times(-i)=-i \).

Answer:

b. 10i

Question 2