5%C3%97(1-i)5.jpeg "(1+i)5×(1-i)5")
Simplifying (1+i)⁵ × (1-i)⁵
This article explores the simplification of the complex number expression (1+i)⁵ × (1-i)⁵. We will leverage the properties of complex numbers and binomial theorem to arrive at a concise solution.
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).
Utilizing the Binomial Theorem
The binomial theorem allows us to expand expressions of the form (x + y)ⁿ. In our case, we can apply it to both (1+i)⁵ and (1-i)⁵:
(1 + i)⁵ = 1⁵ + 5(1⁴)(i) + 10(1³)(i²) + 10(1²)(i³) + 5(1)(i⁴) + i⁵
(1 - i)⁵ = 1⁵ - 5(1⁴)(i) + 10(1³)(i²) - 10(1²)(i³) + 5(1)(i⁴) - i⁵
Simplifying the Expressions
We can simplify the above expressions by remembering that i² = -1, i³ = -i, and i⁴ = 1:
(1 + i)⁵ = 1 + 5i - 10 - 10i + 5 + i = -4 - 4i
(1 - i)⁵ = 1 - 5i - 10 + 10i + 5 - i = -4 + 4i
The Final Calculation
Now, we can multiply the simplified expressions:
(1 + i)⁵ × (1 - i)⁵ = (-4 - 4i) × (-4 + 4i)
This is a product of complex conjugates. The product of complex conjugates simplifies to the sum of the squares of the real and imaginary parts:
(-4 - 4i) × (-4 + 4i) = (-4)² + (4)² = 16 + 16 = 32
Conclusion
Therefore, the simplified form of (1 + i)⁵ × (1 - i)⁵ is 32. This demonstrates how leveraging the properties of complex numbers and binomial theorem allows us to simplify complex expressions into a single real number.