Answer :
To multiply and simplify the product [tex]$(8-5i)^2$[/tex], we can follow these steps:
1. Recognize that the expression is a square, so we write it as:
[tex]$$
(8-5i)^2 = (8-5i)(8-5i)
$$[/tex]
2. Expand the product using the distributive property (also known as the FOIL method):
[tex]$$
(8-5i)(8-5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i)
$$[/tex]
3. Calculate each term:
- First term: [tex]$8 \cdot 8 = 64$[/tex]
- Outer term: [tex]$8 \cdot (-5i) = -40i$[/tex]
- Inner term: [tex]$(-5i) \cdot 8 = -40i$[/tex]
- Last term: [tex]$(-5i) \cdot (-5i) = 25i^2$[/tex]
4. Combine like terms:
[tex]$$
64 - 40i - 40i + 25i^2 = 64 - 80i + 25i^2
$$[/tex]
5. Remember that [tex]$i^2 = -1$[/tex]. Substitute this into the expression:
[tex]$$
25i^2 = 25(-1) = -25
$$[/tex]
6. Now, combine the real parts:
[tex]$$
64 - 25 = 39
$$[/tex]
7. The final simplified result is:
[tex]$$
39 - 80i
$$[/tex]
Thus, the product [tex]$(8-5i)^2$[/tex] simplifies to [tex]$\boxed{39-80i}$[/tex].
1. Recognize that the expression is a square, so we write it as:
[tex]$$
(8-5i)^2 = (8-5i)(8-5i)
$$[/tex]
2. Expand the product using the distributive property (also known as the FOIL method):
[tex]$$
(8-5i)(8-5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i)
$$[/tex]
3. Calculate each term:
- First term: [tex]$8 \cdot 8 = 64$[/tex]
- Outer term: [tex]$8 \cdot (-5i) = -40i$[/tex]
- Inner term: [tex]$(-5i) \cdot 8 = -40i$[/tex]
- Last term: [tex]$(-5i) \cdot (-5i) = 25i^2$[/tex]
4. Combine like terms:
[tex]$$
64 - 40i - 40i + 25i^2 = 64 - 80i + 25i^2
$$[/tex]
5. Remember that [tex]$i^2 = -1$[/tex]. Substitute this into the expression:
[tex]$$
25i^2 = 25(-1) = -25
$$[/tex]
6. Now, combine the real parts:
[tex]$$
64 - 25 = 39
$$[/tex]
7. The final simplified result is:
[tex]$$
39 - 80i
$$[/tex]
Thus, the product [tex]$(8-5i)^2$[/tex] simplifies to [tex]$\boxed{39-80i}$[/tex].
