%5E3.jpeg "(a-4)^3")
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:
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First, expand (a - 4) * (a - 4): (a - 4) * (a - 4) = a(a - 4) - 4(a - 4) = a² - 4a - 4a + 16 = a² - 8a + 16
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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^(n-k) * y^k
where (n choose k) is the binomial coefficient, calculated as n!/(k!(n-k)!).
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.