3.jpeg "(5+n)3")
Expanding (5 + n)³
The expression (5 + n)³ represents the cube of the binomial (5 + n). To expand this expression, we can use the following methods:
Method 1: Using the Binomial Theorem
The Binomial Theorem provides a general formula for expanding expressions of the form (x + y)ⁿ:
(x + y)ⁿ = ∑(n choose k) x^(n-k) y^k
where:
- n choose k represents the binomial coefficient, calculated as n!/(k!(n-k)!).
- k ranges from 0 to n.
Applying this to our expression (5 + n)³, we get:
(5 + n)³ = (3 choose 0) 5³ n⁰ + (3 choose 1) 5² n¹ + (3 choose 2) 5¹ n² + (3 choose 3) 5⁰ n³
Calculating the binomial coefficients and simplifying:
(5 + n)³ = 125 + 75n + 15n² + n³
Method 2: Expanding by Multiplication
We can expand (5 + n)³ by multiplying the expression by itself three times:
(5 + n)³ = (5 + n) * (5 + n) * (5 + n)
First, expand the first two terms:
(5 + n) * (5 + n) = 25 + 10n + n²
Then, multiply this result by (5 + n):
(25 + 10n + n²) * (5 + n) = 125 + 50n + 5n² + 25n + 10n² + n³
Combining like terms, we get:
(5 + n)³ = 125 + 75n + 15n² + n³
Conclusion
Both methods lead to the same result:
(5 + n)³ = 125 + 75n + 15n² + n³
This is the expanded form of the expression (5 + n)³, representing the volume of a cube with side length (5 + n).