Simplifying an Algebraic Expression with Exponents and Division
To simplify the given expression, we follow the rules of exponents and division.
The expression is:
\[ \frac{10x^{5n+4} + 125x^{5n+2}}{3x^{5n+3} - 20x^{5n+1}} \]
Let's break it down step by step:
First, factor out the common factor of x in both numerator and denominator.
In the numerator, the smallest power of x is \(x^{5n+2}\), and in the denominator, the smallest power is \(x^{5n+1}\). We factor these out from both parts:
Numerator factor: \(x^{5n+2}\) (leaving \(10x^2 + 125\))
Denominator factor: \(x^{5n+1}\) (leaving \(3x^2 - 20\))
After factoring out, we have:
\[ \frac{x^{5n+2}(10x^2 + 125)}{x^{5n+1}(3x^2 - 20)} \]
Now we can simplify \(x^{5n+2}\) and \(x^{5n+1}\). Subtracting the exponents (using the properties of exponents), we get \(x^{5n+2 - (5n+1)} = x^{(5n+2) - 5n - 1} = x^1 = x\).
The simplified form of the expression is:
\[ \frac{x(10x^2 + 125)}{3x^2 - 20} \]
We cannot simplify any further without knowing the specific values for x. This is the simplified expression in terms of x and n.
