Simplifying Complex Fraction Expressions Involving Exponents

In this article, we will explore a complex fraction expression involving exponents and learn how to simplify it step by step. We will discuss the given expression and demonstrate multiple methods to reach the final simplified answer. Understanding these techniques is crucial for mastering algebra and simplifying various mathematical expressions.

Introduction to Exponential Expressions and Complex Fractions

Exponential expressions are mathematical notations used to represent repeated multiplication of a base number by itself. Complex fractions, on the other hand, consist of fractions inside a fraction. In this context, we will focus on simplifying a fraction that includes exponential terms in both the numerator and the denominator.

The Given Expression and Simplification Steps

Let's consider the given expression:

[frac{5^{n-4}5^{n-3}5^{n-2}}{5^{n-1}5^n}]

First, let's rewrite the expression:

[frac{5^{n-4}5^{n-3}5^{n-2}}{5^{n-1}5^n}] can be written as:

[frac{5^n5^{-4}5^n5^{-3}5^n5^{-2}}{5^n5^{-1}5^n}] [frac{5^{n-4 n-3 n-2}}{5^{n-1 n}}] [frac{5^{3n-9}}{5^{2n-1}}]

Next, we simplify the exponents in the numerator and the denominator:

[frac{5^{3n-9}}{5^{2n-1}} 5^{(3n-9) - (2n-1)} 5^{3n-9 - 2n 1} 5^{n-8}]

Alternative Methods and Simplifications

Let's explore another method to simplify the same expression:

Divide both the numerator and the denominator by the common factor (5^n):

[frac{5^{n-4}5^{n-3}5^{n-2}}{5^{n-1}5^n} frac{5^{n-4}5^{n-3}5^{n-2}}{5^n5^{n-1}}]

Now, simplify the numerator and the denominator:

Numerator: [5^{n-4}5^{n-3}5^{n-2} 625 cdot 125 cdot 25 775]

Denominator: [5^n5^{n-1} 6]

Thus, the simplified expression is:

[frac{775}{6}]

Solutions and Additional Examples

To further understand the simplification process, let's look at additional examples:

[frac{5^{n-2}5^{n-1}5^n}{5^n5^{n-1}} frac{5^{n-2}5^{n-1}5^n}{5^{2n-1}.5^n} frac{5^{3n-3}}{5^{2n-1}} frac{5^{n-2}}{1} 25 cdot 31 / 6 775/6 129.17]

Let's factorize the numerator and the denominator step by step:

[frac{5^n(5^4)5^n(5^3)5^n(5^2)}{5^n(5^1)5^n} frac{5^n(5^2)5^n(5^2)5^n(5^1)5^n(1)}{5^n(5)^n(5^1)} frac{5^2(25)5^2(25)5^1(5)1}{5^1(5)(1)} 25 cdot 31 / 6 775/6 129.17]

Conclusion

By exploring multiple methods and examples, we have demonstrated how to simplify a complex fraction expression involving exponents. This process enhances our understanding of exponential expressions and complex fractions, providing valuable skills for further mathematical studies and problem-solving in various fields.

Key Takeaways

Understanding exponential notation and complex fractions. Step-by-step methods to simplify exponential expressions. Applying factorization and common factors to simplify complex fractions. Common mistakes and how to avoid them.

Keywords

Exponential Expressions

Exponential expressions are mathematical representations of repeated multiplication of a base number by itself, such as (5^n). They are fundamental in various mathematical and scientific applications.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. This process is essential in algebra and other branches of mathematics, making calculations easier and more manageable.

Complex Fractions

Complex fractions include fractions within fractions, creating a more intricate structure that requires a systematic approach to simplify. Understanding complex fractions helps in solving more complex mathematical problems.