expand (2-i)^12
To expand the expression (2 - i)^12, we can use the Binomial Theorem. The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where a and b are any real or complex numbers, and n is a positive integer.
The formula for the Binomial Theorem is:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
Here, C(n, k) represents the binomial coefficient, also known as "n choose k", which is the number of ways to choose k items from a set of n items.
In our case, we have (2 - i)^12, which means a = 2 and b = -i, and we want to find the expansion up to the 12th power.
Let's now calculate the terms using the Binomial Theorem:
(2 - i)^12 = C(12, 0) * 2^12 * (-i)^0 + C(12, 1) * 2^11 * (-i)^1 + C(12, 2) * 2^10 * (-i)^2 + ... + C(12, 10) * 2^2 * (-i)^10 + C(12, 11) * 2^1 * (-i)^11 + C(12, 12) * 2^0 * (-i)^12
We can simplify the calculations to find the values of the terms:
C(12, 0) = 1
C(12, 1) = 12
C(12, 2) = 66
C(12, 3) = 220
C(12, 4) = 495
C(12, 5) = 792
C(12, 6) = 924
C(12, 7) = 792
C(12, 8) = 495
C(12, 9) = 220
C(12, 10) = 66
C(12, 11) = 12
C(12, 12) = 1
Now, let's substitute these values and simplify further:
(2 - i)^12 = 1 * 2^12 * (-i)^0 + 12 * 2^11 * (-i)^1 + 66 * 2^10 * (-i)^2 + ... + 66 * 2^2 * (-i)^10 + 12 * 2^1 * (-i)^11 + 1 * 2^0 * (-i)^12
Simplifying each term:
(-i)^0 = 1
(-i)^1 = -i
(-i)^2 = -1
(-i)^3 = i
(-i)^4 = 1
(-i)^5 = -i
(-i)^6 = -1
(-i)^7 = i
(-i)^8 = 1
(-i)^9 = -i
(-i)^10 = -1
(-i)^11 = i
(-i)^12 = 1
Substituting these values back into the expression:
(2 - i)^12 = 1 * 2^12 * 1 + 12 * 2^11 * (-i) + 66 * 2^10 * (-1) + ... + 66 * 2^2 * (-1) + 12 * 2^1 * (i) + 1 * 2^0 * (1)
Finally, simplifying further:
(2 - i)^12 = 4096 - 98304i + 98304 - 229376i + 196608 - 196608i + 114688 - 114688i + 40960 - 40960i + 9216 - 9216i + 1024 - 1024i + 4096
Therefore, the expanded form of (2 - i)^12 is:
(2 - i)^12 = 4096 - 98304i + 98304 - 229376i + 196608 - 196608i + 114688 - 114688i + 40960 - 40960i + 9216 - 9216i + 1024 - 1024i + 4096