Determine the 5th term of (2x+3y)^12
To find the 5th term of the expansion (2x+3y)^12, we can use the binomial theorem. The binomial theorem can be applied to expand any expression of the form (a+b)^n, where a and b are variables and n is a positive integer.
The binomial theorem provides a formula for finding the terms of the expanded expression. The formula consists of n+1 terms, with each term having a coefficient and powers of a and b. The coefficients can be determined using combinations, and the powers of a and b change in a pattern.
In this case, we are given (2x+3y)^12, so a = 2x, b = 3y, and n = 12. To find the 5th term, we need to determine the term with the power of a being raised to the (12-5) = 7th power, and the power of b being raised to the 5th power.
Using the binomial theorem, we can write the general term as:
nCr * (a^(n-r)) * (b^r),
where nCr represents the number of combinations of n things taken r at a time.
In our case, the 5th term would be:
12C5 * (2x)^(12-5) * (3y)^5.
Simplifying this expression, we get:
792 * (2x)^7 * (3y)^5.
The 5th term of the expression (2x+3y)^12 is 792 * (2x)^7 * (3y)^5.