Find the product of 9 -5i and its complex conjugate
(9-5i)(9+5i) = 81+25 = 106
note
(a-b)(a+b) = a^2 - b^2
but
i^2 = -1
so
(a-bi)(a+bi) = a^2-b^2i^2 = a^2+b^2
To find the product of a complex number and its complex conjugate, we can apply the formula: (a + bi)(a - bi) = a^2 - b^2i^2.
In this case, a = 9 and b = -5.
Let's substitute these values into the formula:
(9 - 5i)(9 + 5i) = 9^2 - (-5)^2i^2
= 81 - 25i^2
Since i^2 = -1, we can simplify further:
81 - 25i^2 = 81 - 25(-1)
= 81 + 25
= 106
Therefore, the product of 9 - 5i and its complex conjugate is 106.
To find the product of a complex number and its complex conjugate, you can follow these steps:
Step 1: Write down the complex number.
The complex number is 9 - 5i.
Step 2: Write down the complex conjugate.
To find the complex conjugate, you need to change the sign of the imaginary part. In this case, the complex conjugate of 9 - 5i is 9 + 5i.
Step 3: Multiply the complex number by its complex conjugate.
(9 - 5i) * (9 + 5i) = 9 * 9 + 9 * 5i - 5i * 9 - 5i * 5i
Step 4: Simplify the multiplication.
Simplify each term in the product.
= 81 + 45i - 45i - 25i^2
Step 5: Simplify further using the definition of i^2.
Remember that i^2 is defined as -1, so replace i^2 with -1.
= 81 + 45i - 45i - 25(-1)
= 81 + 45i - 45i + 25
= 81 + 25
= 106
Therefore, the product of 9 - 5i and its complex conjugate 9 + 5i is 106.