Understanding (a  4)³
The expression (a  4)³ represents the cube of the binomial (a  4). It means we are multiplying (a  4) by itself three times:
(a  4)³ = (a  4) * (a  4) * (a  4)
There are two main ways to solve this expression:
1. Expanding using the distributive property:
We can use the distributive property to expand the expression step by step:

First, expand (a  4) * (a  4): (a  4) * (a  4) = a(a  4)  4(a  4) = a²  4a  4a + 16 = a²  8a + 16

Now, multiply the result by (a  4): (a²  8a + 16) * (a  4) = a(a²  8a + 16)  4(a²  8a + 16) = a³  8a² + 16a  4a² + 32a  64 = a³  12a² + 48a  64
Therefore, (a  4)³ = a³  12a² + 48a  64.
2. Using the binomial theorem:
The binomial theorem provides a formula for expanding expressions of the form (x + y)ⁿ:
(x + y)ⁿ = ∑(k=0)^n (n choose k) * x^(nk) * y^k
where (n choose k) is the binomial coefficient, calculated as n!/(k!(nk)!).
Applying this to our expression (a  4)³:
 Identify x and y: x = a, y = 4
 Identify n: n = 3
 Expand using the binomial theorem: (a  4)³ = (3 choose 0) * a³ * (4)⁰ + (3 choose 1) * a² * (4)¹ + (3 choose 2) * a¹ * (4)² + (3 choose 3) * a⁰ * (4)³ = 1 * a³ * 1 + 3 * a² * (4) + 3 * a * 16 + 1 * 1 * (64) = a³  12a² + 48a  64
Therefore, we reach the same conclusion: (a  4)³ = a³  12a² + 48a  64.
Both methods lead to the same result, and the choice of which one to use depends on your preference and the complexity of the problem.