(m%2B1)(m%2B1)(m%2B1)(m%2B1).jpeg "(m+1)(m+1)(m+1)(m+1)(m+1)")
Expanding (m+1)(m+1)(m+1)(m+1)(m+1)
This expression represents the product of (m+1) multiplied by itself five times. We can simplify this by using the concept of exponents and binomial expansion.
Understanding Exponents
The expression (m+1)(m+1)(m+1)(m+1)(m+1) can be written in a more compact form using exponents:
(m+1)^5
This means (m+1) multiplied by itself five times.
Binomial Expansion
To expand (m+1)^5, we can use the binomial theorem. The binomial theorem provides a formula for expanding expressions of the form (x+y)^n, where n is a positive integer.
The Binomial Theorem: (x + y)^n = x^n + (n choose 1) x^(n-1) y + (n choose 2) x^(n-2) y^2 + ... + (n choose n-1) xy^(n-1) + y^n
where (n choose k) represents the binomial coefficient, calculated as: (n choose k) = n! / (k! * (n-k)!)
Applying the Binomial Theorem to (m+1)^5
-
Identify x and y: In this case, x = m and y = 1.
-
Apply the formula: (m+1)^5 = m^5 + (5 choose 1) m^4 * 1 + (5 choose 2) m^3 * 1^2 + (5 choose 3) m^2 * 1^3 + (5 choose 4) m * 1^4 + 1^5
-
Calculate the binomial coefficients:
- (5 choose 1) = 5!/(1! * 4!) = 5
- (5 choose 2) = 5!/(2! * 3!) = 10
- (5 choose 3) = 5!/(3! * 2!) = 10
- (5 choose 4) = 5!/(4! * 1!) = 5
- Substitute the coefficients and simplify: (m+1)^5 = m^5 + 5m^4 + 10m^3 + 10m^2 + 5m + 1
Therefore, the expanded form of (m+1)(m+1)(m+1)(m+1)(m+1) is m^5 + 5m^4 + 10m^3 + 10m^2 + 5m + 1.