%5E4.jpeg "(3+i)^4")
Exploring the Power of Complex Numbers: (3 + i)^4
In the realm of mathematics, complex numbers often present unique challenges and fascinating results. One such intriguing exploration involves calculating the power of a complex number, specifically (3 + i)^4. Let's delve into this problem and unravel its solution.
Understanding the Complex Number
Before we embark on the calculation, it's crucial to understand the fundamental nature of complex numbers. A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' represents the imaginary unit, defined as the square root of -1 (i^2 = -1).
Utilizing the Binomial Theorem
The most straightforward approach to calculating (3 + i)^4 is by employing the Binomial Theorem. This theorem states that for any real numbers 'a' and 'b' and a positive integer 'n':
(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k
Where (n choose k) represents the binomial coefficient, calculated as n!/(k!(n-k)!).
Applying this theorem to our problem, we get:
(3 + i)^4 = (4 choose 0) * 3^4 * i^0 + (4 choose 1) * 3^3 * i^1 + (4 choose 2) * 3^2 * i^2 + (4 choose 3) * 3^1 * i^3 + (4 choose 4) * 3^0 * i^4
Simplifying the Expression
Let's simplify each term using the properties of complex numbers:
- i^0 = 1
- i^1 = i
- i^2 = -1
- i^3 = i^2 * i = -i
- i^4 = (i^2)^2 = (-1)^2 = 1
Substituting these values and calculating the binomial coefficients, we obtain:
(3 + i)^4 = 81 + 108i - 54 - 36i + 1
Combining Real and Imaginary Components
Finally, we combine the real and imaginary terms:
**(3 + i)^4 = ** (81 - 54 + 1) + (108 - 36)i = 28 + 72i
Therefore, the solution to (3 + i)^4 is 28 + 72i.
Conclusion
Calculating the power of a complex number like (3 + i)^4 involves utilizing the Binomial Theorem, carefully simplifying the terms, and combining the real and imaginary components. This process demonstrates the elegant structure and rich properties of complex numbers, making them an essential tool in various mathematical and scientific fields.