(8-8i)-in-standard-form.jpeg "(2+3i)(8-8i) In Standard Form")
Simplifying Complex Numbers: (2 + 3i)(8 - 8i)
This article will guide you through the process of simplifying the complex number expression (2 + 3i)(8 - 8i) into standard form (a + bi), where 'a' and 'b' are real numbers.
Understanding Complex Numbers
Complex numbers are expressed in the form a + bi, where:
- a is the real part of the number.
- b is the imaginary part of the number.
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Simplifying the Expression
To simplify the given expression, we'll use the distributive property (also known as FOIL):
- Multiply the first terms: (2)(8) = 16
- Multiply the outer terms: (2)(-8i) = -16i
- Multiply the inner terms: (3i)(8) = 24i
- Multiply the last terms: (3i)(-8i) = -24i²
Now, we have: 16 - 16i + 24i - 24i²
Remember that i² = -1. Substitute this value into the expression:
16 - 16i + 24i - 24(-1)
Simplify:
16 - 16i + 24i + 24
Combine like terms:
40 + 8i
Conclusion
Therefore, the standard form of the complex number (2 + 3i)(8 - 8i) is 40 + 8i.