Write the 6th term of the expansion of

(x^2-2y)^9

and simplify.

Thanks!

http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html

I will be happy to check your work.

Thanks for the link.

But what happens when there is a number in front of one of the variables, like the 2y instead of y?

Thanks again.

Do (a+b)^9 with the formula

a^9 + 9 a^8b + 36 a^7b^2 + 84 a^6b^3 + 126 a^5b^4 ..... etc
Then let a = x^2
and let b= (2y)
so for example b^3 = (2y)^3 = 8y^3

To find the 6th term of the expansion of (x^2-2y)^9, we will use the binomial theorem.

The binomial theorem states that the term at position k in the expansion of (a+b)^n can be found using the formula:

T(k) = (nCk) * a^(n-k) * b^k

Where "nCk" represents the binomial coefficient, which is the number of ways to choose k items from n items. The formula to calculate the binomial coefficient is:

nCk = n! / (k! * (n-k)!)

In the given expression, (x^2-2y)^9, our "a" will be x^2, "b" will be -2y, and "n" will be 9.

Using the binomial theorem, the 6th term will be:
T(6) = (9C6) * (x^2)^(9-6) * (-2y)^6

Let's calculate each part of the equation:

First, calculate 9C6:
9C6 = 9! / (6! * (9-6)!)
= 9! / (6! * 3!)
= (9 * 8 * 7) / (3 * 2 * 1)
= 84

Next, calculate (x^2)^(9-6):
(x^2)^(9-6) = x^2^3
= x^6

Finally, calculate (-2y)^6:
(-2y)^6 = (-2)^6 * (y^6)
= 64 * y^6

Now, substitute the values back into the formula:
T(6) = (9C6) * (x^2)^(9-6) * (-2y)^6
= 84 * (x^6) * (64 * y^6)

To simplify, multiply the coefficients and combine like terms:
T(6) = 84 * 64 * x^6 * y^6
= 537,600x^6y^6

Therefore, the 6th term of the expansion of (x^2-2y)^9 is 537,600x^6y^6.