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The expression you need to simplify is:

\[
\frac{7 + i}{7 - i}
\]

To write the quotient in standard form (a + bi), we need to multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(7 - i\) is \(7 + i\).

### Step-by-step:

1. Multiply both the numerator and denominator by \(7 + i\):
   \[
   \frac{7 + i}{7 - i} \times \frac{7 + i}{7 + i} = \frac{(7 + i)(7 + i)}{(7 - i)(7 + i)}
   \]

2. Simplify the denominator:
   \[
   (7 - i)(7 + i) = 7^2 - i^2 = 49 - (-1) = 50
   \]

3. Simplify the numerator by expanding \((7 + i)(7 + i)\):
   \[
   (7 + i)(7 + i) = 7^2 + 2(7)(i) + i^2 = 49 + 14i + (-1) = 48 + 14i
   \]

4. Now the expression becomes:
   \[
   \frac{48 + 14i}{50}
   \]

5. Divide both the real and imaginary parts by 50:
   \[
   \frac{48}{50} + \frac{14i}{50} = 0.96 + 0.28i
   \]

So the simplified answer in standard form is:
\[
0.96 + 0.28i
\]

Would you like further details on any of the steps?

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Here are some related questions you might want to explore:
1. What is a complex conjugate and why is it used in division of complex numbers?
2. How do you multiply two complex numbers?
3. What is the significance of \(i^2 = -1\) in complex number calculations?
4. How do you convert complex numbers from polar to rectangular form?
5. What is the geometric interpretation of dividing complex numbers?

**Tip**: Always remember that multiplying by the conjugate helps eliminate the imaginary part in the denominator.

Write the quotient in standard form (Simplify your answer completely): (7 + i) / (7 - i)

Learn how to simplify the quotient of complex numbers (7 + i) / (7 - i) and express it in standard form (a + bi). This step-by-step solution includes multiplying by the complex conjugate and dividing the real and imaginary parts. Suitable for high school students in Grades 10-12.

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