Question

simplify.

\\(-8 + 5i)^2\\

Explanation:

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<reasoning>

Identify the expression and expansion formula

We need to simplify the square of a complex binomial:
\[ (-8 + 5i)^2 \]
We will use the binomial expansion formula:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
where \(a = -8\) and \(b = 5i\).

Apply the binomial expansion formula

Substitute \(a = -8\) and \(b = 5i\) into the formula:
\[ (-8 + 5i)^2 = (-8)^2 + 2(-8)(5i) + (5i)^2 \]

Simplify each term individually

Calculate the value of each term in the expansion:
\[ (-8)^2 = 64 \]
\[ 2(-8)(5i) = -80i \]
\[ (5i)^2 = 25i^2 \]

Substitute the value of the imaginary unit

Recall that the imaginary unit \(i\) satisfies:
\[ i^2 = -1 \]
Substitute this into the third term:
\[ 25i^2 = 25(-1) = -25 \]

Combine the real and imaginary parts

Combine the real numbers and keep the imaginary term:
\[ 64 - 80i - 25 = (64 - 25) - 80i = 39 - 80i \]
</reasoning>

<answer>
\(39 - 80i\)
</answer>

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Answer:

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<reasoning>

Identify the expression and expansion formula

We need to simplify the square of a complex binomial:
\[ (-8 + 5i)^2 \]
We will use the binomial expansion formula:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
where \(a = -8\) and \(b = 5i\).

Apply the binomial expansion formula

Substitute \(a = -8\) and \(b = 5i\) into the formula:
\[ (-8 + 5i)^2 = (-8)^2 + 2(-8)(5i) + (5i)^2 \]

Simplify each term individually

Calculate the value of each term in the expansion:
\[ (-8)^2 = 64 \]
\[ 2(-8)(5i) = -80i \]
\[ (5i)^2 = 25i^2 \]

Substitute the value of the imaginary unit

Recall that the imaginary unit \(i\) satisfies:
\[ i^2 = -1 \]
Substitute this into the third term:
\[ 25i^2 = 25(-1) = -25 \]

Combine the real and imaginary parts

Combine the real numbers and keep the imaginary term:
\[ 64 - 80i - 25 = (64 - 25) - 80i = 39 - 80i \]
</reasoning>

<answer>
\(39 - 80i\)
</answer>

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