Write the quotient in standard form. (Simplify your answer completely.) 4/5 − 8i

To write the quotient \( \frac{4}{5 - 8i} \) in standard form (i.e., \( a + bi \), where \( a \) and \( b \) are real numbers), follow these steps:

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of \( 5 - 8i \) is \( 5 + 8i \).

\[
\frac{4}{5 - 8i} \cdot \frac{5 + 8i}{5 + 8i} = \frac{4(5 + 8i)}{(5 - 8i)(5 + 8i)}
\]

### Step 2: Simplify the denominator using the difference of squares formula.

\[
(5 - 8i)(5 + 8i) = 5^2 - (8i)^2 = 25 - (-64) = 25 + 64 = 89
\]

So the denominator becomes 89.

### Step 3: Distribute in the numerator.

\[
4(5 + 8i) = 4 \cdot 5 + 4 \cdot 8i = 20 + 32i
\]

### Step 4: Combine the result.

\[
\frac{20 + 32i}{89} = \frac{20}{89} + \frac{32}{89}i
\]

### Final Answer:

\[
\frac{4}{5 - 8i} = \frac{20}{89} + \frac{32}{89}i
\]

This is the simplified quotient in standard form.

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Do you want more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate eliminate the imaginary part from the denominator?
3. What is the difference of squares formula and how is it used in complex number division?
4. Can you explain how to represent a complex number graphically?
5. What are the real and imaginary parts of the complex number \( \frac{20}{89} + \frac{32}{89}i \)?

**Tip:** Always multiply by the conjugate of the denominator when dividing by complex numbers to simplify them.

Learn how to simplify the quotient 4/(5 - 8i) and express it in standard form (a + bi). This solution includes multiplying by the conjugate, using the difference of squares formula, and distributing terms. Ideal for Grades 10-12 students learning about complex numbers.

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