Answer :
To solve the problem of multiplying and simplifying the product [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Start with the given expression:
[tex]\[(8 - 5i)^2\][/tex]
2. Use the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
In this expression, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Substitute the values into the formula:
[tex]\[(8)^2 - 2(8)(5i) + (5i)^2\][/tex]
4. Calculate each term:
- The first term is [tex]\((8)^2 = 64\)[/tex].
- The second term is [tex]\(-2(8)(5i) = -80i\)[/tex].
- The third term is [tex]\((5i)^2 = 25i^2\)[/tex].
5. Recall that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25(-1) = -25\)[/tex].
6. Substitute these results back:
[tex]\[64 - 80i - 25\][/tex]
7. Combine the real parts:
[tex]\[64 - 25 = 39\][/tex]
8. Write the final simplified expression:
[tex]\[39 - 80i\][/tex]
Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
1. Start with the given expression:
[tex]\[(8 - 5i)^2\][/tex]
2. Use the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
In this expression, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Substitute the values into the formula:
[tex]\[(8)^2 - 2(8)(5i) + (5i)^2\][/tex]
4. Calculate each term:
- The first term is [tex]\((8)^2 = 64\)[/tex].
- The second term is [tex]\(-2(8)(5i) = -80i\)[/tex].
- The third term is [tex]\((5i)^2 = 25i^2\)[/tex].
5. Recall that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25(-1) = -25\)[/tex].
6. Substitute these results back:
[tex]\[64 - 80i - 25\][/tex]
7. Combine the real parts:
[tex]\[64 - 25 = 39\][/tex]
8. Write the final simplified expression:
[tex]\[39 - 80i\][/tex]
Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
