Simplifying the Expression: 5*25^n+1-25*5^2 n/5*5^2n+3-25^n+1
In this article, we will simplify the given expression: 5*25^n+1-25*5^2 n/5*5^2n+3-25^n+1. This expression involves exponential functions, multiplication, and division, which can make it seem complex at first glance. However, by applying the rules of exponents and simplifying the expression step by step, we can arrive at a much simpler form.
Step 1: Simplify the Exponential Terms
The first step is to simplify the exponential terms in the expression. Recall that 25 can be written as 5^2, so we can rewrite the expression as:
5*(5^2)^n+1-25*5^2 n/5*5^2n+3-(5^2)^n+1
Using the rule of exponents that states a^(mn) = (a^m)^n, we can rewrite the expression as:
5*5^(2n)+1-25*5^2 n/5*5^2n+3-5^(2n)+1
Step 2: Simplify the Fractions
Next, we need to simplify the fraction term in the expression. To do this, we can rewrite the fraction as:
25*5^2 n / 5*5^2n = 25n / 5^(2n)
Using the rule of exponents that states a^m / a^n = a^(m-n), we can rewrite the fraction as:
25n / 5^(2n) = 5^(2-2n)n = 5^(2-2n)
Step 3: Combine Like Terms
Now, we can combine the like terms in the expression. We have two terms with 5^(2n) and two terms with 1, so we can combine them as:
5*5^(2n)+1-5^(2-2n)-5^(2n)+1
Step 4: Simplify the Expression
Finally, we can simplify the expression by combining the like terms:
5^(2n)(5-1)+2 = 4*5^(2n)+2
And that's the simplified expression!
Conclusion
In this article, we have simplified the expression 5*25^n+1-25*5^2 n/5*5^2n+3-25^n+1 by applying the rules of exponents and combining like terms. The simplified expression is 4*5^(2n)+2.
