%5E4.jpeg "(2m-1)^4")
Expanding (2m-1)^4
The expression (2m-1)^4 represents the fourth power of the binomial (2m-1). To expand this, we can use the binomial theorem or simply multiply the expression by itself four times.
Using the Binomial Theorem
The binomial theorem states that:
(a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where:
- n is the power
- k ranges from 0 to n
- (n choose k) is the binomial coefficient, calculated as n!/(k! * (n-k)!)
Applying this to our problem:
- a = 2m
- b = -1
- n = 4
Therefore, we have:
(2m - 1)^4 = Σ (4 choose k) * (2m)^(4-k) * (-1)^k
This results in the following expansion:
(4 choose 0) * (2m)^4 * (-1)^0 + (4 choose 1) * (2m)^3 * (-1)^1 + (4 choose 2) * (2m)^2 * (-1)^2 + (4 choose 3) * (2m)^1 * (-1)^3 + (4 choose 4) * (2m)^0 * (-1)^4
Simplifying the binomial coefficients and powers:
1 * 16m^4 * 1 + 4 * 8m^3 * -1 + 6 * 4m^2 * 1 + 4 * 2m * -1 + 1 * 1 * 1
Finally, we get the expanded form:
**(2m - 1)^4 = ** 16m^4 - 32m^3 + 24m^2 - 8m + 1
Expanding by Multiplication
We can also expand (2m-1)^4 by multiplying the expression by itself four times:
(2m - 1)^4 = (2m - 1) * (2m - 1) * (2m - 1) * (2m - 1)
First, multiply the first two factors:
(2m - 1) * (2m - 1) = 4m^2 - 4m + 1
Then, multiply this result by the third factor:
(4m^2 - 4m + 1) * (2m - 1) = 8m^3 - 12m^2 + 6m - 1
Finally, multiply this result by the last factor:
(8m^3 - 12m^2 + 6m - 1) * (2m - 1) = 16m^4 - 32m^3 + 24m^2 - 8m + 1
As we can see, both methods lead to the same expanded form: 16m^4 - 32m^3 + 24m^2 - 8m + 1.