(-4-7i)(6-5i).jpeg "(1+3i)(-4-7i)(6-5i)")
Multiplying Complex Numbers: (1+3i)(-4-7i)(6-5i)
This article will guide you through the process of multiplying three complex numbers: (1+3i)(-4-7i)(6-5i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplying Complex Numbers
To multiply complex numbers, we distribute just like with regular binomials.
Step 1: Multiply the first two complex numbers
(1+3i)(-4-7i)
- Distribute:
- 1 * (-4) + 1 * (-7i) + 3i * (-4) + 3i * (-7i)
- Simplify:
- -4 -7i - 12i -21i²
- Substitute i² = -1
- -4 -7i - 12i + 21
- Combine real and imaginary terms:
- 17 -19i
Step 2: Multiply the result by the third complex number
(17 - 19i)(6 - 5i)
- Distribute:
- 17 * 6 + 17 * (-5i) + (-19i) * 6 + (-19i) * (-5i)
- Simplify:
- 102 - 85i - 114i + 95i²
- Substitute i² = -1
- 102 - 85i - 114i - 95
- Combine real and imaginary terms:
- 7 - 199i
Final Result
Therefore, the product of (1+3i)(-4-7i)(6-5i) is 7 - 199i.