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Chapter 10: Problem 567

Simplify. $$\frac{x}{x^{5}}$$

Short Answer

Expert verified
x^{-4}

Step by step solution

01

Understand the Problem

The goal is to simplify the fraction \(\frac{x}{x^5}\). This means expressing it in its simplest form.
02

Apply the Laws of Exponents

Recall that \( \frac{a^m}{a^n} = a^{m-n} \). Here, both the numerator and the denominator have the same base, \ (x) \, so the formula applies.
03

Rewrite the Expression

Using the laws of exponents, rewrite the expression: \( \frac{x}{x^5} = x^{1-5} \).
04

Simplify the Exponent

Simplify the exponent: \( x^{1-5} = x^{-4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

laws of exponents
Understanding the laws of exponents is essential when simplifying expressions, especially fractions.
The main rule we used in this exercise is: \( \frac{a^m}{a^n} = a^{m-n} \).
This means that when you divide two powers that have the same base, you simply subtract the exponent in the denominator from the exponent in the numerator.
Remember, \(aeq 0\) to ensure the division is valid.
simplify fractions
To simplify fractions involving exponents, identify common bases in the numerator and denominator.
In the exercise, we have \( \frac{x}{x^5} \), where both the numerator and the denominator share the base \( x \).
Using the laws of exponents, we can rewrite the fraction to simplify it.
By applying \( \frac{a^m}{a^n} = a^{m-n} \), we get \( x^{1-5} = x^{-4} \).
Simplification is about making expressions shorter and cleaner.
negative exponents
Negative exponents can be a bit tricky, but they follow the same rules as positive exponents.
A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.
In other words, \( a^{-n} = \frac{1}{a^n} \).
In our simplified expression \( x^{-4} \), we can convert it to \( \frac{1}{x^4} \) if needed.
Understanding this concept helps in switching between different forms of the expression easily.

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