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.