Question

Prove that (1+I)^4into (1+1/i)^4=16

Answer 1

Given

(1+i)⁴ × (1+1/i)⁴

To Prove

(1+i)⁴ × (1+1/i)⁴ = 16

By using Complex identity

1/i = -i

We can write as

(1 + i)⁴ × (1 - i)⁴

{(1 + i)²}² × {(1 - i)²}²

Using this identities

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

We get

{ (1)² + (i)² + 2i }² × { (1)² + (i)² - 2i }

{ 1 - 1 + 2i }² × { 1 - 1 - 2i }²

{ 0 + 2i }² x { - 2i }²

{2i}² × {-2i}²

(2)²(i)² × (-2)²(i)²

4 × - 1 × 4 × -1

-4×-4

16

Hence Proved

More Information

i² = -1

i³ = -i

i⁴ = 1

1/i = -1

Answer 2

Question:-

• Prove that ( 1 + i )⁴ × (1 + 1 / i )⁴ = 16

Solution:-

Here ,

️ ➭ ( 1 + i )⁴ × (1 + 1 / i )⁴ = 16

Can also be written as : -

️ ➭ [ ( 1 + i )² ]² × [ ( 1 - i )² ]²

Now ,

️ ➭ ( 1² + i² + 2i )² × ( 1² + i² - 2i )²

️ ➭ ( 1 - 1 + 2i )² × ( 1 - 1 - 2i )²

️ ➭ ( 2 )² ( i )² × ( - 2 )² ( i )²

️ ➭ 4 × - 1 × 4 × - 1

️ ➭ 16

• Hence Proved