%5E5.jpeg "(3a-4b)^5")
Expanding (3a - 4b)^5
The expression (3a - 4b)^5 represents the product of (3a - 4b) multiplied by itself five times. Expanding this expression can be done using the Binomial Theorem.
The Binomial Theorem
The Binomial Theorem provides a formula to expand expressions of the form (x + y)^n, where n is a non-negative integer. The formula is:
(x + y)^n = x^n + (n choose 1)x^(n-1)y + (n choose 2)x^(n-2)y^2 + ... + (n choose n-1)xy^(n-1) + y^n
where (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Binomial Theorem
To expand (3a - 4b)^5, we can substitute x = 3a, y = -4b, and n = 5 into the Binomial Theorem formula.
This gives us:
(3a - 4b)^5 = (3a)^5 + (5 choose 1)(3a)^4(-4b) + (5 choose 2)(3a)^3(-4b)^2 + (5 choose 3)(3a)^2(-4b)^3 + (5 choose 4)(3a)(-4b)^4 + (-4b)^5
Now, we can calculate the binomial coefficients and simplify the expression:
- (5 choose 1) = 5! / (1! * 4!) = 5
- (5 choose 2) = 5! / (2! * 3!) = 10
- (5 choose 3) = 5! / (3! * 2!) = 10
- (5 choose 4) = 5! / (4! * 1!) = 5
Substituting these values and simplifying, we get:
(3a - 4b)^5 = 243a^5 - 1620a^4b + 4320a^3b^2 - 5760a^2b^3 + 3840ab^4 - 1024b^5
Conclusion
Therefore, the expanded form of (3a - 4b)^5 is 243a^5 - 1620a^4b + 4320a^3b^2 - 5760a^2b^3 + 3840ab^4 - 1024b^5. The Binomial Theorem provides a systematic way to expand any expression of the form (x + y)^n.