Answer :
To simplify the product
[tex]$$
(8-5i)^2,
$$[/tex]
follow these steps:
1. Write the square as a product of two factors:
[tex]$$
(8-5i)^2 = (8-5i)(8-5i).
$$[/tex]
2. Expand by applying the formula for the square of a binomial:
[tex]$$
(8-5i)^2 = 8^2 + 2\cdot8\cdot(-5i) + (-5i)^2.
$$[/tex]
3. Calculate each term:
- The square of the real part:
[tex]$$
8^2 = 64.
$$[/tex]
- Twice the product of the real and imaginary parts:
[tex]$$
2\cdot8\cdot(-5i) = -80i.
$$[/tex]
- The square of the imaginary part:
[tex]$$
(-5i)^2 = (-5)^2\cdot i^2 = 25\cdot(-1) = -25 \quad \text{(since } i^2 = -1\text{)}.
$$[/tex]
4. Combine all the computed terms:
[tex]$$
64 - 80i - 25.
$$[/tex]
5. Simplify by combining the real numbers:
[tex]$$
64 - 25 = 39,
$$[/tex]
so the expression becomes:
[tex]$$
39 - 80i.
$$[/tex]
Thus, the simplified product is
[tex]$$
\boxed{39-80i}.
$$[/tex]
[tex]$$
(8-5i)^2,
$$[/tex]
follow these steps:
1. Write the square as a product of two factors:
[tex]$$
(8-5i)^2 = (8-5i)(8-5i).
$$[/tex]
2. Expand by applying the formula for the square of a binomial:
[tex]$$
(8-5i)^2 = 8^2 + 2\cdot8\cdot(-5i) + (-5i)^2.
$$[/tex]
3. Calculate each term:
- The square of the real part:
[tex]$$
8^2 = 64.
$$[/tex]
- Twice the product of the real and imaginary parts:
[tex]$$
2\cdot8\cdot(-5i) = -80i.
$$[/tex]
- The square of the imaginary part:
[tex]$$
(-5i)^2 = (-5)^2\cdot i^2 = 25\cdot(-1) = -25 \quad \text{(since } i^2 = -1\text{)}.
$$[/tex]
4. Combine all the computed terms:
[tex]$$
64 - 80i - 25.
$$[/tex]
5. Simplify by combining the real numbers:
[tex]$$
64 - 25 = 39,
$$[/tex]
so the expression becomes:
[tex]$$
39 - 80i.
$$[/tex]
Thus, the simplified product is
[tex]$$
\boxed{39-80i}.
$$[/tex]
