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Chapter 9: Problem 33

Divide as indicated. Write your answer using only positive exponents. $$\frac{x^{3}}{x^{4}}$$

Short Answer

Expert verified
\( \frac{1}{x} \)

Step by step solution

01

Understanding the Problem

Here, we need to divide \( x^3 \) by \( x^4 \). Division of exponential terms with the same base involves subtracting the exponents.
02

Apply the Division Property of Exponents

When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. For our problem, this is \( x^{3 - 4} \).
03

Simplify the Expression

Perform the subtraction: \( 3 - 4 = -1 \). So, we have \( x^{-1} \).
04

Write the Answer with Positive Exponents

To express \( x^{-1} \) with a positive exponent, we take the reciprocal: \( x^{-1} = \frac{1}{x} \).

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Key Concepts

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

Division of Exponents
When you come across dividing two exponent expressions that share the same base, there's a consistent shortcut you can always use. It's all about subtraction. Simply subtract the exponent in the denominator from the exponent in the numerator. This rule makes it so much easier to handle big expressions without directly multiplying out all the numbers. For instance, let's look at \( \frac{x^3}{x^4} \). Both the numerator and the denominator have the base \( x \). According to the division rule, we subtract the exponents: \( 3 - 4 \) gives us \( x^{-1} \) as the result. Remember:
  • If you divide equal bases, subtract the exponents: \( x^a \div x^b = x^{a-b} \).
  • The base remains the same; only the exponents get subtracted.
This principle works on all similar expressions with common bases.
Positive Exponents
Dealing with negative exponents isn't as tricky as it seems. When you encounter negative exponents, it's your cue to think of reciprocals. A negative exponent tells you to "flip" the base to the other side of a fraction bar. Let's consider \( x^{-1} \). To convert it into a positive exponent, you move the base to the denominator, making it \( \frac{1}{x} \). By doing this, you've changed the exponent from negative to positive. Encouraging notes on positive exponents:
  • Negative exponents indicate a fraction: \( x^{-b} = \frac{1}{x^b} \).
  • Always aim to rewrite any negative exponents as positive ones, especially when finalizing your answers.
Positive exponents give a clearer, friendlier presentation of your solution.
Simplifying Expressions
Simplifying expressions is all about finding that concise, clean way to represent your mathematical work. It means making expressions as straightforward and easy to read as possible, without changing their value. In our example \( \frac{x^3}{x^4} \), the initial simplification gets us to \( x^{-1} \). However, because we strive for clarity, we take that further by writing it as \( \frac{1}{x} \). This form is considered more polished because it uses positive exponents. Here are some simplification tips:
  • Always look to express with positive exponents.
  • Simplify step by step and ensure each step maintains the original value of the expression.
  • Break down complex steps into small, manageable parts.
Simplifying makes your work shine and helps anyone who reads it to understand it with ease.

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