Write the conjugate of ( 2 + 3 i ) ( 1 - 2 i ) in the form a + i b , a , b ∈ R .

Let z = ( 2 + 3 i ) ( 1 - 2 i ) ⇒ z = 2 - 4 i + 3 i - 6 i 2 ⇒ z = 2 - i - 6 i 2 ⇒ z = 2 - i + 6 ∵ i 2 = - 1 ⇒ z = 8 - i We know that, the conjugate of x + i y is x - i y . Hence, the conjugate of 8 - i is 8 + i .

Write the conjugate of ( 2 + 3 i ) ( 1 - 2 i ) in the form a + i b , a , b ∈ R .

Let z = ( 2 + 3 i ) ( 1 - 2 i ) ⇒ z = 2 - 4 i + 3 i - 6 i 2 ⇒ z = 2 - i - 6 i 2 ⇒ z = 2 - i + 6 ∵ i 2 = - 1 ⇒ z = 8 - i We know that, the conjugate of x + i y is x - i y . Hence, the conjugate of 8 - i is 8 + 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 .