Which question could be the result of two complex conjugates being multiplied together
16i
20
9-9i
25+4i
What is the product of the complex conjugates 16i and -16i?
What is the answer?
The product of the complex conjugates 16i and -16i is 256.
The question that could be the result of two complex conjugates being multiplied together is:
"What is the product of (a+bi) and (a-bi)?"
To find the product of two complex conjugates, you can use the FOIL method, which stands for First, Outer, Inner, Last. Here's how to apply it:
Step 1: Identify the values of a and b from the expressions you have provided.
For example, let's use the expression (a+bi)(a-bi).
From the given expressions:
- For "16i," a = 0 and b = 16.
- For "20," a = 20 and b = 0.
- For "9-9i," a = 9 and b = -9.
- For "25+4i," a = 25 and b = 4.
Step 2: Apply the FOIL method to multiply the two complex conjugates.
(a+bi)(a-bi) = a^2 - abi + abi - (bi)^2
Simplifying the expression further:
- (bi)^2 = -b^2i^2 (using the property i^2 = -1)
- -b^2i^2 = -b^2(-1)
- -b^2(-1) = b^2
After simplification:
(a+bi)(a-bi) = a^2 - b^2
Therefore, the result of multiplying two complex conjugates is a^2 - b^2.
Substituting the values of a and b from the original expressions you provided, we can calculate the results for each case:
- For "16i": a = 0, b = 16
Result: (0)^2 - (16)^2 = -256
- For "20": a = 20, b = 0
Result: (20)^2 - (0)^2 = 400
- For "9-9i": a = 9, b = -9
Result: (9)^2 - (-9)^2 = 18 - 81 = -63
- For "25+4i": a = 25, b = 4
Result: (25)^2 - (4)^2 = 625 - 16 = 609
Therefore, the products of the given expressions are:
- 16i: -256
- 20: 400
- 9-9i: -63
- 25+4i: 609