%5E5.jpeg "(2a+2b)^5")
Expanding (2a + 2b)^5 using the Binomial Theorem
The binomial theorem provides a powerful way to expand expressions of the form (x + y)^n. Let's apply it to expand (2a + 2b)^5.
The Binomial Theorem
The binomial theorem states:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
where:
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
- ∑_(k=0)^n represents the sum from k = 0 to n.
Expanding (2a + 2b)^5
- Identify x and y: In our case, x = 2a and y = 2b.
- Identify n: n = 5.
Now, let's apply the binomial theorem:
(2a + 2b)^5 = ∑_(k=0)^5 (5 choose k) * (2a)^(5-k) * (2b)^k
Let's calculate each term individually:
- k = 0: (5 choose 0) * (2a)^5 * (2b)^0 = 1 * 32a^5 * 1 = 32a^5
- k = 1: (5 choose 1) * (2a)^4 * (2b)^1 = 5 * 16a^4 * 2b = 160a^4b
- k = 2: (5 choose 2) * (2a)^3 * (2b)^2 = 10 * 8a^3 * 4b^2 = 320a^3b^2
- k = 3: (5 choose 3) * (2a)^2 * (2b)^3 = 10 * 4a^2 * 8b^3 = 320a^2b^3
- k = 4: (5 choose 4) * (2a)^1 * (2b)^4 = 5 * 2a * 16b^4 = 160ab^4
- k = 5: (5 choose 5) * (2a)^0 * (2b)^5 = 1 * 1 * 32b^5 = 32b^5
Therefore, the expanded form of (2a + 2b)^5 is:
(2a + 2b)^5 = 32a^5 + 160a^4b + 320a^3b^2 + 320a^2b^3 + 160ab^4 + 32b^5