Find each product. Write answers in standard form. (6 – 4i)(6 + 4i)
The Correct Answer and Explanation is:
Correct Answer:
(6 – 4i)(6 + 4i) = 52
Explanation:
To find the product of two complex numbers, especially when they are in the form (a – bi)(a + bi), you can recognize this as a difference of squares problem. The difference of squares formula is:
(a – b)(a + b) = a² – b²
In this case:
- a = 6
- b = 4i
So applying the formula:
(6 – 4i)(6 + 4i) = 6² – (4i)²
Now calculate each part:
- 6² = 36
- (4i)² = 16i²
Recall that i² = -1, because i is the imaginary unit defined by √(-1). So:
16i² = 16(-1) = -16
Now substitute:
36 – (-16) = 36 + 16 = 52
So, the product of (6 – 4i)(6 + 4i) is 52, a real number.
This result demonstrates an important concept in complex numbers. When a complex number is multiplied by its conjugate, the imaginary parts cancel out, and the product is always a real number. This is useful when simplifying fractions that contain complex numbers in the denominator because multiplying by the conjugate helps eliminate the imaginary unit from the denominator.
This method is widely used in algebra and complex arithmetic to simplify expressions and rationalize denominators. It also shows how imaginary numbers interact with real numbers when squared and multiplied. Recognizing conjugate pairs and using the difference of squares shortcut saves time and reduces calculation errors.
