To express \( \frac{6 + i}{(5 - 5i)^2} \) in standard form, we need to follow these steps:

### Step 1: Simplify the denominator
First, we calculate \( (5 - 5i)^2 \). To do this, use the formula for squaring a binomial:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
Let \( a = 5 \) and \( b = 5i \). Now, apply the formula:
\[
(5 - 5i)^2 = 5^2 - 2(5)(5i) + (5i)^2
\]
\[
= 25 - 50i + 25i^2
\]
Since \( i^2 = -1 \), this simplifies to:
\[
25 - 50i + 25(-1) = 25 - 50i - 25 = -50i
\]
Thus, \( (5 - 5i)^2 = -50i \).

### Step 2: Multiply numerator and denominator by the conjugate of the denominator
Now we have:
\[
\frac{6 + i}{-50i}
\]
To simplify this, multiply both the numerator and denominator by the conjugate of the denominator, which is \( i \) (since \( i \cdot i = -1 \)).

\[
\frac{(6 + i) \cdot i}{-50i \cdot i} = \frac{(6 + i)i}{-50(-1)}
\]

### Step 3: Simplify
First, distribute \( i \) in the numerator:
\[
(6 + i)i = 6i + i^2 = 6i - 1
\]
So the expression becomes:
\[
\frac{6i - 1}{50} = \frac{-1}{50} + \frac{6i}{50}
\]

### Step 4: Write in standard form
The standard form of a complex number is \( a + bi \), where \( a \) and \( b \) are real numbers. Therefore, the final expression is:
\[
\frac{-1}{50} + \frac{6i}{50} = -\frac{1}{50} + \frac{3i}{25}
\]

Thus, the standard form of \( \frac{6 + i}{(5 - 5i)^2} \) is:
\[
-\frac{1}{50} + \frac{3i}{25}
\]

Would you like more details on any step or have questions?

### Related questions:
1. How do you find the conjugate of a complex number?
2. What is the general formula for dividing complex numbers?
3. How can you simplify powers of \( i \)?
4. What are the properties of binomials involving complex numbers?
5. How do you convert a complex number from polar to rectangular form?

**Tip:** When dividing complex numbers, multiplying by the conjugate of the denominator often simplifies the process.

Learn how to simplify the complex expression 6 + i / (5 - 5i)^2 into standard form. This step-by-step solution covers key concepts such as complex numbers, binomial expansion, and the conjugate method. Suitable for students in Grades 10-12.

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