%5E4.jpeg "(5-5i)^4")
Simplifying (5 - 5i)^4
This article will guide you through the process of simplifying the complex number (5 - 5i)^4.
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 (i.e., i² = -1).
Utilizing the Binomial Theorem
To simplify (5 - 5i)^4, we can utilize the binomial theorem:
(a + b)^n = a^n + (n choose 1) a^(n-1)b + (n choose 2) a^(n-2)b^2 + ... + (n choose n-1) ab^(n-1) + b^n
where (n choose k) represents the binomial coefficient, calculated as n! / (k! (n-k)!).
Applying the Binomial Theorem
- Identify a and b: In our case, a = 5 and b = -5i.
- Expand using the binomial theorem: (5 - 5i)^4 = 5^4 + (4 choose 1) 5^3 (-5i) + (4 choose 2) 5^2 (-5i)^2 + (4 choose 3) 5 (-5i)^3 + (-5i)^4
- Simplify each term:
- 5^4 = 625
- (4 choose 1) 5^3 (-5i) = 4 * 125 * (-5i) = -2500i
- (4 choose 2) 5^2 (-5i)^2 = 6 * 25 * 25 = 3750
- (4 choose 3) 5 (-5i)^3 = 4 * 5 * (-125i) = -2500i
- (-5i)^4 = 625
Combining Terms
Now, combine the simplified terms: (5 - 5i)^4 = 625 - 2500i + 3750 - 2500i + 625 = 7500 - 5000i
Conclusion
Therefore, the simplified form of (5 - 5i)^4 is 7500 - 5000i.