Express 1 3 + 7 i + ( 9 - 2 i ) - - 4 3 + 2 i in the form a + i b .

Given the expression is : 1 3 + 7 i + ( 9 - 2 i ) - - 4 3 + 2 i = 1 3 + 7 i + 9 - 2 i + 4 3 - 2 i = 1 3 + 9 + 4 3 + i 7 - 2 - 2 = 5 3 + 9 + i 3 = 32 3 + 3 i = a + i b , where a = 32 3 and b = 3 .

Find the conjugate of 1 2 - 7 i .

Let a + i b = 1 2 - 7 i = 2 + 7 i 2 - 7 i 2 + 7 i = 2 + 7 i 2 2 - 7 i 2 = 2 + 7 i 4 - 49 i 2 = 2 + 7 i 4 - 49 - 1 = 2 + 7 i 53 So, the conjugate of a + i b is a - i b = 2 - 7 i 53 .

Find the principal argument of 2 3 - 2 i .

Let z = 2 3 - 2 i Let 2 3 = r cos θ and - 2 = r sin θ By squaring and adding, we get ( 2 3 ) 2 + ( - 2 ) 2 = ( r cos θ ) 2 + ( r sin θ ) 2 ⇒ 12 + 4 = r 2 cos 2 θ + sin 2 θ ⇒ 16 = r 2 ⇒ r = 4 ∴ cos θ = 3 2 and sin θ = - 1 2 ⇒ tan θ = - 1 3 Since, θ lies in fourth quadrant, we have principal argument = - - π 6 = π 6 . Since, θ ∈ ( - π , π ] it is principal argument.

Find the value of ( - 3 ) ( - 6 ) .

Since, i = - 1 ⇒ - 3 - 6 = - 1 3 - 1 6 = i 2 3 6 = - 1 × 18 = - 3 2

Find the value of i 272 + i 270 + i 268 + i 266 + i 264 i 282 + i 280 + i 278 + i 276 + i 274 .

Let z = i 272 + i 270 + i 268 + i 266 + i 264 i 282 + i 280 + i 278 + i 276 + i 274 = i 264 1 + i 2 + i 4 + i 6 + i 8 i 274 1 + i 2 + i 4 + i 6 + i 8 = i 264 i 274 = 1 i 274 - 264 = 1 i 10 = 1 i 4 × 2 + 2 = 1 i 4 2 i 2 = 1 1 2 - 1 = - 1 .

Express 1 2 + 2 i + 2 3 - i - - 2 5 + i in the form a + i b .

Given the expression is : 1 2 + 2 i + ( 2 3 - i ) - - 2 5 + i = 1 2 + 2 i + 2 3 - i + 2 5 - i = 1 2 + 2 3 + 2 5 + i 2 - 1 - 1 = 15 + 20 + 12 30 + i 0 = 47 30 + i 0 = a + i b , where a = 47 30 and b = 0 .

Evaluate - 2 - 1 4 i 3 .

Let x = - 2 - 1 4 i 3 = - 1 3 2 + i 4 3 = - 2 3 + 3 2 2 i 4 + 3 2 i 4 2 + i 4 3 = - 8 + 3 i + 3 - 1 8 - i 64 = - 61 8 + 191 i 64 = - 61 8 - 191 i 64 .