%5E5-expansion.jpeg "(e-1)^5 Expansion")
Expansion of (e-1)^5
In this article, we will explore the expansion of (e-1)^5, where e is the base of the natural logarithm. This expansion is an important concept in mathematics, particularly in calculus and number theory.
What is e?
Before we dive into the expansion of (e-1)^5, let's briefly discuss what e is. e is a mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is a fundamental constant in mathematics, appearing in many mathematical formulas.
Binomial Theorem
To expand (e-1)^5, we can use the binomial theorem, which states that:
$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$
where n is a positive integer, and binom(n,k) is the binomial coefficient.
Expansion of (e-1)^5
Using the binomial theorem, we can expand (e-1)^5 as follows:
$(e-1)^5 = \sum_{k=0}^5 \binom{5}{k} e^{5-k} (-1)^k$
Expanding the summation, we get:
$(e-1)^5 = e^5 - 5e^4 + 10e^3 - 10e^2 + 5e - 1$
Simplification
We can simplify the expansion by combining like terms:
$(e-1)^5 = e^5 - 5e^4 + 10e^3 - 10e^2 + 5e - 1$
Conclusion
In conclusion, the expansion of (e-1)^5 is a fundamental concept in mathematics, and using the binomial theorem, we can expand it as e^5 - 5e^4 + 10e^3 - 10e^2 + 5e - 1. This expansion has many applications in calculus, number theory, and other areas of mathematics.
