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Multiplying Complex Numbers: (7 + 8i)(7 - 8i)
This article will guide you through the process of multiplying the complex numbers (7 + 8i) and (7 - 8i).
Understanding Complex Numbers
A complex number is a number 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 use the distributive property, just like we do with regular binomials.
Step 1: Expand the product using the distributive property (also known as FOIL):
(7 + 8i)(7 - 8i) = 7(7) + 7(-8i) + 8i(7) + 8i(-8i)
Step 2: Simplify the terms:
= 49 - 56i + 56i - 64i²
Step 3: Substitute i² with -1:
= 49 - 56i + 56i - 64(-1)
Step 4: Combine like terms:
= 49 + 64 = 113
The Result
Therefore, the product of (7 + 8i) and (7 - 8i) is 113.
Important Note:
The result is a real number. This is because (7 + 8i) and (7 - 8i) are complex conjugates of each other. Complex conjugates are pairs of complex numbers that differ only in the sign of their imaginary part. When complex conjugates are multiplied, the imaginary terms cancel out, leaving a real number result.