Write the quotient in standard form.
 
11
3 − i
 

To write the quotient \(\frac{11}{3 - i}\) in standard form, we need to eliminate the imaginary unit \(i\) from the denominator. This can be done by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of \(3 - i\) is \(3 + i\).

### Step-by-step process:

1. Multiply both the numerator and denominator by the conjugate of the denominator:

\[
\frac{11}{3 - i} \cdot \frac{3 + i}{3 + i} = \frac{11(3 + i)}{(3 - i)(3 + i)}
\]

2. Expand the numerator:

\[
11(3 + i) = 11 \cdot 3 + 11 \cdot i = 33 + 11i
\]

So the numerator becomes \(33 + 11i\).

3. Now expand the denominator:

\[
(3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10
\]

4. Now the quotient becomes:

\[
\frac{33 + 11i}{10}
\]

5. Divide both the real and imaginary parts by 10:

\[
\frac{33}{10} + \frac{11}{10}i = 3.3 + 1.1i
\]

### Final Answer:

The quotient in standard form is:

\[
3.3 + 1.1i
\]

Do you want further details or have any questions? Here are 5 related questions for further exploration:

1. How do you find the conjugate of a complex number?
2. Why is multiplying by the conjugate important in complex division?
3. What is the geometric interpretation of dividing complex numbers?
4. Can the result of division of complex numbers ever be purely real or purely imaginary?
5. How would the quotient change if the denominator had a different imaginary part?

**Tip:** Always multiply both numerator and denominator by the conjugate when dividing complex numbers to remove the imaginary part from the denominator.

Write the quotient in standard form.

\(\frac{11}{3 - i}\)

Learn how to write the quotient of 11 divided by (3 - i) in standard form with a step-by-step solution. Explore key concepts of complex numbers, including using the conjugate for division. Suitable for students in Grades 9-12.

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