How do I expand these powers?
(a - b/2)^4
(2- m/2)^5
To expand the given powers, you can use the binomial theorem. The binomial theorem states that the expansion of (a + b)^n can be expressed using the binomial coefficients and the powers of a and b. The formula for the binomial theorem is:
(a + b)^n = (n choose 0) * a^n * b^0 + (n choose 1) * a^(n-1) * b^1 + (n choose 2) * a^(n-2) * b^2 + ... + (n choose k) * a^(n-k) * b^k + ... + (n choose n) * a^0 * b^n
Let's use this formula to expand the given powers:
1. (a - b/2)^4:
Here, a = a and b/2 = b. The power is 4.
Using the binomial theorem, the expansion will be:
(4 choose 0) * a^4 * b^0 + (4 choose 1) * a^3 * b^1 + (4 choose 2) * a^2 * b^2 + (4 choose 3) * a^1 * b^3 + (4 choose 4) * a^0 * b^4
Simplifying this expression, we get:
a^4 - 4a^3(b/2) + 6a^2(b/2)^2 - 4a(b/2)^3 + (b/2)^4
Further simplifying:
a^4 - 2a^3b + 3a^2b^2/4 - ab^3/2 + b^4/16
Therefore, the expanded form of (a - b/2)^4 is a^4 - 2a^3b + (3/4)a^2b^2 - (1/2)ab^3 + (1/16)b^4.
2. (2 - m/2)^5:
Here, 2 = a and m/2 = b. The power is 5.
Using the binomial theorem, the expansion will be:
(5 choose 0) * 2^5 * (m/2)^0 + (5 choose 1) * 2^4 * (m/2)^1 + (5 choose 2) * 2^3 * (m/2)^2 + (5 choose 3) * 2^2 * (m/2)^3 + (5 choose 4) * 2^1 * (m/2)^4 + (5 choose 5) * 2^0 * (m/2)^5
Simplifying this expression, we get:
2^5 - 5(2^4)(m/2) + 10(2^3)(m/2)^2 - 10(2^2)(m/2)^3 + 5(2^1)(m/2)^4 - (m/2)^5
Further simplifying:
32 - 40m + 20m^2 - 5m^3 + (m^4)/8
Therefore, the expanded form of (2 - m/2)^5 is 32 - 40m + 20m^2 - 5m^3 + (1/8)m^4.
So, the expanded forms are:
(a - b/2)^4 = a^4 - 2a^3b + (3/4)a^2b^2 - (1/2)ab^3 + (1/16)b^4
(2 - m/2)^5 = 32 - 40m + 20m^2 - 5m^3 + (1/8)m^4