%5E3.jpeg "(4n−3)^3")
Expanding the Expression (4n - 3)³
The expression (4n - 3)³ represents the cube of the binomial (4n - 3). This means we are multiplying the binomial by itself three times:
(4n - 3)³ = (4n - 3) * (4n - 3) * (4n - 3)
To expand this expression, we can use the following methods:
Method 1: Repeated Multiplication
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Multiply the first two binomials: (4n - 3) * (4n - 3) = 16n² - 12n - 12n + 9 = 16n² - 24n + 9
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Multiply the result by the remaining binomial: (16n² - 24n + 9) * (4n - 3) = 64n³ - 96n² + 36n - 48n² + 72n - 27
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Combine like terms: 64n³ - 144n² + 108n - 27
Method 2: Binomial Theorem
The binomial theorem provides a formula to expand expressions of the form (a + b)ⁿ:
(a + b)ⁿ = ∑(n choose k) * a^(n-k) * b^k
Where:
- ∑ represents the sum from k = 0 to n.
- (n choose k) = n! / (k! * (n-k)!) is the binomial coefficient.
Applying this to (4n - 3)³:
- a = 4n, b = -3, and n = 3
Therefore, the expansion becomes:
(4n - 3)³ = (3 choose 0) * (4n)³ * (-3)⁰ + (3 choose 1) * (4n)² * (-3)¹ + (3 choose 2) * (4n)¹ * (-3)² + (3 choose 3) * (4n)⁰ * (-3)³
Calculating the binomial coefficients and simplifying:
(4n - 3)³ = 1 * 64n³ * 1 + 3 * 16n² * -3 + 3 * 4n * 9 + 1 * 1 * -27
Finally:
(4n - 3)³ = 64n³ - 144n² + 108n - 27
Conclusion
Both methods lead to the same result: (4n - 3)³ = 64n³ - 144n² + 108n - 27. Understanding the methods helps to expand similar expressions and apply the concept of binomial theorem for various applications.