Understanding the Simplification of nth Roots and Exponential Expressions
In this article, we will explore a specific algebraic manipulation involving nth roots and exponential expressions. We aim to understand the process of simplifying the expression (frac{sqrt[n]{x^{n-1}}}{x}) and show how it can be reduced to (sqrt[n]{x}).
Step-by-Step Illustration of the Process
Let's break down the problem (frac{sqrt[n]{x^{n-1}}}{x}). Our goal is to show that this expression can be simplified to (sqrt[n]{x}).
Step 1: Introduce the nth Root in the Numerator
First, we rewrite the numerator using the properties of nth roots:
[frac{sqrt[n]{x^{n-1}}}{x}]Next, we recognize that the nth root of a product can be split into the product of the nth roots:
[frac{sqrt[n]{x^{n-1}}}{x} frac{sqrt[n]{x cdot x^{n-2}}}{x}]Further simplifying, we can separate the terms under the root:
[ frac{sqrt[n]{x} cdot sqrt[n]{x^{n-2}}}{x}]Step 2: Simplify the Exponential Expression
Now, let's simplify the exponential expression in the numerator:
[frac{sqrt[n]{x} cdot x^{frac{n-2}{n}}}{x}]This step involves recognizing that (sqrt[n]{x} x^{frac{1}{n}}):
[ frac{x^{frac{1}{n}} cdot x^{frac{n-2}{n}}}{x}]Using the property of exponents that states (x^a cdot x^b x^{a b}), we combine the exponents in the numerator:
[ frac{x^{frac{1}{n} frac{n-2}{n}}}{x}]Next, we simplify the exponent in the numerator:
[ frac{x^{frac{1 n - 2}{n}}}{x}] [ frac{x^{frac{n-1}{n}}}{x}]Step 3: Final Simplification
Using the property (frac{x^a}{x^b} x^{a-b}), we simplify the expression:
[ x^{frac{n-1}{n} - 1}]Note that 1 can be written as (frac{n}{n}):
[ x^{frac{n-1}{n} - frac{n}{n}}] [ x^{frac{n-1-n}{n}}] [ x^{frac{-1}{n}}]This is the reciprocal of the nth root of (x):
[ frac{1}{sqrt[n]{x}}]However, recognizing that the original expression (frac{sqrt[n]{x^{n-1}}}{x}) represents the nth root of (x^{n-1}) divided by (x), we have:
[ sqrt[n]{x}]Hence, the simplified form of (frac{sqrt[n]{x^{n-1}}}{x}) is indeed (sqrt[n]{x}).
Conclusion
In conclusion, we have successfully simplified the expression (frac{sqrt[n]{x^{n-1}}}{x}) to (sqrt[n]{x}) using properties of nth roots and exponents. This process highlights the importance of recognizing and applying these fundamental algebraic properties.
Further Reading and Practice
For a deeper understanding, consider exploring more complex algebraic expressions and their simplifications. Practice with similar problems and seek out additional resources such as textbooks and online tutorials to enhance your skills in simplifying and manipulating exponential and root expressions.