%5E6-6(m-n)%5E4%2B12(m-b)%5E2-8.jpeg "(m-n)^6-6(m-n)^4+12(m-b)^2-8")
Expanding and Simplifying (m-n)^6 - 6(m-n)^4 + 12(m-n)^2 - 8
This expression appears complex, but we can simplify it using the properties of exponents and algebraic manipulation. Let's break it down step-by-step:
1. Recognize the Pattern
Notice the pattern in the exponents: 6, 4, 2, and 0 (implied in the constant term -8). This suggests that the expression might be a result of expanding a binomial raised to a power.
2. Consider the Binomial Theorem
The binomial theorem helps us expand expressions of the form (x + y)^n. In this case, our binomial is (m-n) and the powers are 6, 4, 2, and 0.
3. Apply the Binomial Theorem
Using the binomial theorem, we can write:
(m - n)^6 = m^6 - 6m^5n + 15m^4n^2 - 20m^3n^3 + 15m^2n^4 - 6mn^5 + n^6
(m - n)^4 = m^4 - 4m^3n + 6m^2n^2 - 4mn^3 + n^4
(m - n)^2 = m^2 - 2mn + n^2
4. Substitute and Simplify
Now, substitute these expanded expressions back into the original expression:
(m^6 - 6m^5n + 15m^4n^2 - 20m^3n^3 + 15m^2n^4 - 6mn^5 + n^6) - 6(m^4 - 4m^3n + 6m^2n^2 - 4mn^3 + n^4) + 12(m^2 - 2mn + n^2) - 8
Distribute the constants and combine like terms:
m^6 - 6m^5n + 15m^4n^2 - 20m^3n^3 + 15m^2n^4 - 6mn^5 + n^6 - 6m^4 + 24m^3n - 36m^2n^2 + 24mn^3 - 6n^4 + 12m^2 - 24mn + 12n^2 - 8
After simplifying, we get the final expanded form:
m^6 - 6m^5n + 9m^4n^2 - 4m^3n^3 + 3m^2n^4 - 6mn^5 + n^6 - 6m^4 + 24m^3n - 24m^2n^2 + 24mn^3 - 6n^4 + 12m^2 - 24mn + 12n^2 - 8
Conclusion
This simplified expression represents the fully expanded form of the original expression. While it appears complex, understanding the binomial theorem and applying algebraic manipulation allows us to break it down into manageable steps.