Question

Two complex numbers are being divided as shown below.

16-3i
2 + 14i
Part A:
What is the complex conjugate of the denominator?
Part B:
What is the quotient of these complex numbers written in standard (a + bi)?

Answer 1

Part A: The complex conjugate is
`\(2 - 14i\).`

Part B: The final answer in standard form (a + bi) is
`\[ -(1)/(20) - (23)/(20)i \]`

How did we get the values?

Given complex numbers:

`\( (16 - 3i)/(2 + 14i) \)`

Part A: Complex Conjugate of the Denominator

The complex conjugate of a complex number
`\(a + bi\)` is
`\(a - bi\).` For the given denominator
`\(2 + 14i\)`, the complex conjugate is
`\(2 - 14i\).`

Part B: Quotient in Standard Form (a + bi)

To find the quotient, we need to multiply the given expression by the conjugate of the denominator over itself. This is done to eliminate the imaginary part from the denominator.

`\[ (16 - 3i)/(2 + 14i) \cdot (2 - 14i)/(2 - 14i) \]`

Now, multiply the numerators and denominators:

Numerator:
`\( (16 - 3i)(2 - 14i) = 32 - 224i - 6i + 42i^2 \)`

Simplify:
`\( 32 - 230i - 42 \)` (because
`\(i^2 = -1\))`

Combine like terms:
`\( -10 - 230i \)`

Denominator:
`\( (2 + 14i)(2 - 14i) = 4 - 196i^2 \)`

Simplify:
`\( 4 + 196 \)` (because
`\(i^2 = -1\)`)

Combine like terms:
`\( 200 \)`

Now, the quotient is:

`\[ (-10 - 230i)/(200) \]`

Simplify the fraction by dividing both the real and imaginary parts by 10:

`\[ (-1 - 23i)/(20) \]`

So, the final answer in standard form (a + bi) is:

`\[ -(1)/(20) - (23)/(20)i \]`