For Q.1 to Q.10 express each complex number in form of a+ib

Question 1. (5i)(-3i/5)

Solution:

Let the given number be a,

a= (5i)*(-3i/5)

a= (-15i²)/5

a= (-3)*i²

a= (-3)*(-1)

a= 3+0i

Question 2. i⁹+i¹⁹

Solution:

Let the given number be a,

a = i⁹ * (1+i¹⁰)

a = ((i⁴)²*i )(1 + (i⁴)² (i²))

a = (1*i)(1+i²)

a = (i)*(0)

a = 0+0i

Question 3. i^-39

Solution:

Let the given number be a and let z = i³⁹ ,

z = (i)*(i²)¹⁹

z = (i)*(-1)¹⁹

z = -i

a = i^-39 

a = 1/i³⁹

a = 1/z

a = 1/-i

a = (i⁴)/-i

a = -i³ = -(i²*i)

a = -1*-i

a = 0+i

Question 4. 3(7+7i) + i(7+7i)

Solution:

Let the given number be a,

a = 3*(7+7i)+i*(7+7i)

a = 21+21i+7i+7i²

a = 21+7i²+28i

a = 21-7+28i

a = 14+28i          

Question 5. (1-i)-(-1+i6)

Solution:

Let the given number be a,

a = (1-i)-(-1+6i)

a = 1-i+1-6i

a = 2-7i         

Question 6. (1/5+2i/5)-(4+5i/2)

Solution:

Let the given number be a,

a = (1/5+2i/5)-(4+5i/2)

a = (1/5-4)+(2i/5-5i/2)

a = (-19/5)+(2/5-5/2)i

a = -19/5+(-21/10)i

a = (-38-21i)/10 

Question 7. [(1/3+7i/3)+(4+i/3)]-(-4/3+i)

Solution:

Let the given number be a,

a = (1/3+7i/3)+(4+i/3)-(-4/3+i)

a = (1/3+4+4/3)+(7i/3+i/3-i)

a = (5/3+4)+(8i/3-i)

a = 17/3+ (5i/3)

a = (17+5i)/3

Question 8. (1-i)⁴

Solution:

Let the given number be a,

a = ((1-i)²)²                               

As we know , (a-b)²= (a²+b²-2ab)                                                                                                                                   

a = (1+i²-2i)²

a = (1-1-2i)²

a = (-2i)²

a = 4i²

a = -4+0i

Question 9. (1/3+3i)³

Solution:

Let the given number be a,

a = (1/3+3i)³                              

As we know, (a+b)³= (a³+b³+3ab(a+b))                                                                                                                                  

a = ((1/27)+(3i)³ +3(1/3)*(3i)(1/3+3i))

a = (1/27 +(-27i)+ 3i*(1/3+3i))

a = (1/27+(-27i)+i+9i²)

a = ((1/27)-9+(-27)i+i)

a = ((-242/27)-26i)

Question 10. (-2-(i/3))³

Solution:

Let the given number be a,

a = (-2-i/3)³       

a = -((2+i/3)³)                      

As we know, (a+b)³= (a³+b³+3ab(a+b))                                                                                                                                  

a = -((8)+(i/3)³ +3(2)*(i/3)(2+i/3))

a = -(8+(-i/27)+ 2i*(2+i/3))

a = -(8-i/27+4i+2i²/3) 

a = -(8-2/3+(-i/27)+4i)

a = -(22/3 +(107i/27))

a = -22/3-107i/27