%5E5-(1-i)%5E7.jpeg "(1+i)^5 (1-i)^7")
Simplifying Complex Expressions: (1+i)^5 (1-i)^7
This article will explore the simplification of the complex expression (1+i)^5 (1-i)^7. We'll use the properties of complex numbers and the binomial theorem to arrive at the solution.
Understanding Complex Numbers
Complex numbers are 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 (i.e., i² = -1).
Simplifying (1+i)^5 and (1-i)^7
We can use the binomial theorem to expand these terms. The binomial theorem states:
(a + b)^n = Σ(n choose k) a^(n-k) b^k
Where:
- n is the power
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!)
Applying this to (1+i)^5, we get:
(1 + i)^5 = (5 choose 0) 1^5 i^0 + (5 choose 1) 1^4 i^1 + (5 choose 2) 1^3 i^2 + (5 choose 3) 1^2 i^3 + (5 choose 4) 1^1 i^4 + (5 choose 5) 1^0 i^5
Simplifying this expression, we get:
(1 + i)^5 = 1 + 5i - 10 - 10i + 5 + i = -4 - 4i
Similarly, for (1-i)^7:
(1 - i)^7 = (7 choose 0) 1^7 (-i)^0 + (7 choose 1) 1^6 (-i)^1 + (7 choose 2) 1^5 (-i)^2 + (7 choose 3) 1^4 (-i)^3 + (7 choose 4) 1^3 (-i)^4 + (7 choose 5) 1^2 (-i)^5 + (7 choose 6) 1^1 (-i)^6 + (7 choose 7) 1^0 (-i)^7
Simplifying this expression, we get:
(1 - i)^7 = 1 - 7i + 21 - 35i + 35 - 21i + 7i - 1 = 8 - 56i
Combining the Results
Now, we can combine the simplified expressions of (1+i)^5 and (1-i)^7:
(1+i)^5 (1-i)^7 = (-4 - 4i) (8 - 56i)
Expanding this product:
= -32 + 224i - 32i + 224i²
= -32 + 224i - 32i - 224
= -256 + 192i
Therefore, the simplified form of (1+i)^5 (1-i)^7 is -256 + 192i.