Question

Divide and simplify to the form a+bi. (16i)/(1+i)

Answer 1

`((16i)/(1 + i) )( (1 - i)/(1 - i)) = (16 + 16i)/(2) = 8 + 8i`

Answer 2

`\( (16i)/(1+i) = -8 + 8i \)`

Step-by-step explanation:

To simplify
`\( (16i)/(1+i) \) to the form \( a+bi \)`, we employ a common technique called multiplying by the conjugate. The conjugate of
`\(1+i\) is \(1-i\).`Multiply both the numerator and denominator by
`\(1-i\):`

`\[ (16i)/(1+i) \cdot (1-i)/(1-i) \]`

This simplifies to:

`\[ ((16i)(1-i))/((1+i)(1-i)) \]`

Now, perform the multiplication in the numerator and denominator:

`\[ (16i - 16i^2)/(1 - i^2) \]`

Since
`\(i^2 = -1\)`, substitute this in:

`\[ (16i + 16)/(2) \]`

This further simplifies to:

`\[ 8 + 8i \]`

Thus,
`\( (16i)/(1+i) \) is in the form \( a+bi \),` and the final answer is
`\( -8 + 8i \).` The process of multiplying by the conjugate is a common technique in complex number arithmetic, used to eliminate imaginary units from the denominator and simplify expressions to a standard form.