What is the sixth term of the binomial expansion for (2x−2y)^6
The formula for the sixth term of the binomial expansion is:
(nCk) * a^(n-k) * b^k
In this case, n = 6, a = 2x, b = -2y, and we want to find the term when k = 5. Plugging in these values, we have:
(6C5) * (2x)^(6-5) * (-2y)^5
Simplifying:
6 * (2x)^1 * (-2y)^5
= 6 * 2x * (-2y)^5
= 6 * 2x * (-32y^5)
= -384x * y^5
So, the sixth term of the binomial expansion for (2x−2y)^6 is -384x * y^5.
To find the sixth term of the binomial expansion for (2x−2y)^6, we can use the binomial theorem.
The binomial theorem states that the kth term of the binomial expansion for (a+b)^n is given by the formula:
T(k) = C(n, k) * a^(n-k) * b^k
where C(n, k) represents the binomial coefficient, which is calculated using the formula:
C(n, k) = n! / (k! * (n - k)!)
In this case, we want to find the sixth term, so k is equal to 6.
T(6) = C(6, 6) * (2x)^(6-6) * (-2y)^6
First, let's calculate the binomial coefficient:
C(6, 6) = 6! / (6! * (6 - 6)!) = 1
Now, let's substitute the values into the formula:
T(6) = 1 * (2x)^(6-6) * (-2y)^6
Simplifying further:
T(6) = 1 * (2x)^0 * (-2y)^6
Since any number raised to the power of 0 is 1, we can simplify even more:
T(6) = 1 * 1 * (-2y)^6
Finally:
T(6) = (-2y)^6
The sixth term of the binomial expansion for (2x−2y)^6 is (-2y)^6.
To find the sixth term of the binomial expansion for (2x−2y)^6, you can use the formula for the general term of a binomial expansion. The general term is given by:
T(r+1) = nCr * a^(n-r) * b^r
Where:
T(r+1) is the term number (r+1),
n is the exponent of the binomial (in this case, 6),
r is the term number minus 1 (in this case, 6-1 = 5),
nCr is the binomial coefficient, also known as "n choose r",
a is the coefficient of the first term (in this case, 2x),
b is the coefficient of the second term (in this case, -2y).
Using this formula, we can calculate the sixth term as follows:
nCr = 6C5 = 6! / (5!(6-5)!) = 6
T(6) = 6C5 * (2x)^(6-5) * (-2y)^5
= 6 * 2^(6-5) * x^(6-5) * (-2)^5 * y^5
= 6 * 2 * x * (-32) * y^5
= -384xy^5
Therefore, the sixth term of the binomial expansion for (2x−2y)^6 is -384xy^5.