Question

"Use the definition, identities, and properties of imaginary numbers to simplify each of the following expressions. Match each item with its correct simplest form.

1. - i6 · i3 · i-5 -1
2. ( i3) 5 -i
3. ( i5 · i-2 · i8) 0 1
4. -( i4) ( i0) ( i7) i"

Answer 1

To simplify the expressions using the definition, identities, and properties of imaginary numbers, we can follow the rules for multiplying and raising imaginary numbers to a power. Applying these rules, expression 1 simplifies to -1, expression 2 simplifies to i^15 - i^3, expression 3 simplifies to i^11, and expression 4 simplifies to -i^11.

Step-by-step explanation:

To simplify these expressions using the definition, identities, and properties of imaginary numbers, we can apply the following rules:

1. 
   
3. When multiplying imaginary numbers, $i^m \cdot i^n = i^{m+n}$.
   For example, $i^3 \cdot i^5 = i^{3+5} = i^8$
4. 
   
6. When raising an imaginary number to a power, $(i^n)^m = i^{n \cdot m}$
   For example, $(i^3)^5 = i^{3 \cdot 5} = i^{15}$
7. 
   
9. $i^{4k} = 1$, where $k$ is any integer.
   For example, $i^{4 \cdot 2} = i^8 = 1$
10. 
    
12. $i^0 =1$

Now let's simplify each expression:

1. 
   
3. $-i^6 \cdot i^3 \cdot i^{-5} = -i^{6+3-5} = -i^4 = -1$
4. 
   
6. $(i^3)^5 - i^3 = i^{3 \cdot 5} - i^3 = i^{15} - i^3$
7. 
   
9. $i^5 \cdot i^{-2} \cdot i^8 = i^{5+(-2)+8} = i^{11}$
10. 
    
12. $-i^4 \cdot i^0 \cdot i^7 = -i^{4+0+7} = -i^{11}$