%5E6.jpeg "(m+n)^6")
Expanding (m+n)^6
The expression (m+n)^6 represents the expansion of the binomial (m+n) multiplied by itself six times. Expanding this expression can be done using the Binomial Theorem, which provides a formula for expanding any power of a binomial.
The Binomial Theorem
The Binomial Theorem states:
(m + n)^n = ∑ (n choose k) * m^(n-k) * n^k
Where:
- n is the power to which the binomial is raised
- k is an integer ranging from 0 to n
- (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items. It is calculated as: (n choose k) = n! / (k! * (n-k)!)
Expanding (m+n)^6
Applying the Binomial Theorem to (m+n)^6, we get:
(m + n)^6 = (6 choose 0) * m^6 * n^0 + (6 choose 1) * m^5 * n^1 + (6 choose 2) * m^4 * n^2 + (6 choose 3) * m^3 * n^3 + (6 choose 4) * m^2 * n^4 + (6 choose 5) * m^1 * n^5 + (6 choose 6) * m^0 * n^6
Let's calculate each term:
- (6 choose 0) = 1
- (6 choose 1) = 6
- (6 choose 2) = 15
- (6 choose 3) = 20
- (6 choose 4) = 15
- (6 choose 5) = 6
- (6 choose 6) = 1
Substituting these values, we get:
(m + n)^6 = 1 * m^6 * n^0 + 6 * m^5 * n^1 + 15 * m^4 * n^2 + 20 * m^3 * n^3 + 15 * m^2 * n^4 + 6 * m^1 * n^5 + 1 * m^0 * n^6
Simplifying the expression, we obtain the final expanded form:
(m + n)^6 = m^6 + 6m^5n + 15m^4n^2 + 20m^3n^3 + 15m^2n^4 + 6mn^5 + n^6
Conclusion
By utilizing the Binomial Theorem, we can effectively expand the expression (m+n)^6 into a sum of seven terms. This expansion allows us to understand the relationship between the variables m and n when raised to the power of six. The coefficients in the expansion follow the pattern of Pascal's Triangle, which provides a visual representation of the binomial coefficients.