Express the complex number i + i 2 + i 4 1 + i 2 + i 4 , in standard form a + i b .

Let z = i + i 2 + i 4 1 + i 2 + i 4 = i - 1 + 1 1 - 1 + 1 ∵ i 4 = 1 , i 2 = - 1 = i Hence, the standard form is 0 + i .

Express the complex number i + i 2 + i 4 1 + i 2 + i 4 , in standard form a + i b .

Let z = i + i 2 + i 4 1 + i 2 + i 4 = i - 1 + 1 1 - 1 + 1 ∵ i 4 = 1 , i 2 = - 1 = i Hence, the standard form is 0 + i .

Express the complex number i 2 + i 4 + i 6 + i 7 1 + i 2 + i 3 , in standard form a + i b .

Let z = i 2 + i 4 + i 6 + i 7 1 + i 2 + i 3 = i 2 + i 4 + i 4 × i 2 + i 4 × i 3 1 + i 2 + i 3 = i 2 + 1 + i 2 + i 3 1 + i 2 + i 3 ∵ i 4 = 1 = - 1 + 1 - 1 - i 1 - 1 - i ∵ i 3 = - i , i 2 = - 1 = - 1 - i - i = 1 + i i = 1 + i i × i i = i + i 2 i 2 = i - 1 - 1 ∵ i 2 = - 1 = 1 - i Hence, the standard form is 1 - i .

Express the complex number 1 + i 4 1 + 1 i 4 , in the standard form a + i b .

Let z = 1 + i 4 1 + 1 i 4 ⇒ z = 1 + i 4 1 + 1 i × i i 4 ⇒ z = 1 + i 4 1 + i i 2 4 ⇒ z = 1 + i 4 1 - i 4 ∵ i 2 = - 1 ⇒ z = 1 + i 1 - i 4 ⇒ z = 1 - i 2 4 ⇒ z = 1 + 1 4 ⇒ z = 16 Hence, the standard form is 16 + i . 0 .

Find the multiplicative inverse of 3 + 7 i .

Let z = 3 + 7 i Since we know the multiplicative inverse of z is 1 z ⇒ z - 1 = 1 z = 1 3 + 7 i × 3 - 7 i 3 - 7 i (multiply and divide by conjugate of denominator) = ( 3 - 7 i ) 9 - 7 i 2 = ( 3 - 7 i ) 9 + 7 ∵ i 2 = - 1 = ( 3 - 7 i ) 16 Hence, the multiplicative inverse of 3 + 7 i is 3 16 - 7 16 i .

Express the complex number 5 + 9 i - 3 + 4 i , in the standard form a + i b .

Let z = 5 + 9 i - 3 + 4 i = 5 + 9 i - 3 + 4 i × ( - 3 - 4 i ) ( - 3 - 4 i ) (multiply and divide by conjugate of denominator) = - 15 - 20 i - 27 i - 36 i 2 9 - 16 i 2 = - 15 - 47 i + 36 9 + 16 ∵ i 2 = - 1 = 21 - 47 i 25 So, the standard form is 21 25 - 47 25 i .

Express the complex number ( - 1 + 3 i ) - 1 , in the standard form a + i b .

Let z = ( - 1 + 3 i ) - 1 ⇒ z = 1 ( - 1 + 3 i ) ∵ z - 1 = 1 z = 1 ( - 1 + 3 i ) × ( - 1 - 3 i ) ( - 1 - 3 i ) (multiply and divide by conjugate of denominator) = ( - 1 - 3 i ) ( 1 - 3 i 2 ) = ( - 1 - 3 i ) ( 1 + 3 ) ∵ i 2 = - 1 = ( - 1 - 3 i ) 4 So, the standard form is - 1 4 - 3 4 i .

Express the complex number ( 2 + 3 i ) 2 2 + i , in the standard form a + i b .

Let z = ( 2 + 3 i ) 2 2 + i = 4 + 9 i 2 + 12 i 2 + i = 4 - 9 + 12 i 2 + i = - 5 + 12 i 2 + i = - 5 + 12 i 2 + i × 2 - i 2 - i (multiply and divide by conjugate of denominator) = - 10 + 5 i + 24 i - 12 i 2 4 - i 2 = - 10 + 5 i + 24 i + 12 4 + 1 ∵ i 2 = - 1 = 2 + 29 i 5 So, the standard form is 2 5 + 29 5 i .