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Chapter 10: Problem 381

In the following exercises, simplify. $$\frac{r^{6}}{r^{9}}$$

Short Answer

Expert verified
\(\frac{r^{6}}{r^{9}} = \frac{1}{r^3}\)

Step by step solution

01

Understand the Division of Exponents Rule

The basic rule for dividing exponents with the same base is \(\frac{a^m}{a^n} = a^{m-n}\). This means you subtract the exponent of the denominator from the exponent of the numerator.
02

Apply the Rule to the Given Expression

In the given expression \(\frac{r^6}{r^9}\), the base is \(r\), the exponent in the numerator is 6, and the exponent in the denominator is 9.
03

Subtract the Exponents

Subtract the exponent of the denominator from the exponent of the numerator: \(6 - 9 = -3\). This gives us \(r^{-3}\).
04

Simplify \(r^{-3}\)

An exponent of \(-n\) can be written as the reciprocal with a positive exponent: \(r^{-3} = \frac{1}{r^3}\).

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Key Concepts

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

division of exponents
When working with exponents, one of the essential rules to understand is the division of exponents. When you divide two expressions that have the same base, you can subtract the exponent of the denominator from the exponent of the numerator. This is expressed by the formula: \[ \frac{a^m}{a^n} = a^{m-n} \] The base, denoted as 'a', must be the same for this rule to apply. For example, in the expression \[ \frac{r^6}{r^9} \], the base is 'r', and you subtract the exponent in the denominator (9) from the exponent in the numerator (6). This means: \[ 6 - 9 = -3 \] So, \[ \frac{r^6}{r^9} = r^{-3} \]
negative exponents
Negative exponents can seem tricky at first, but they follow a simple rule. A negative exponent signifies that the base is on the wrong side of a fraction line and needs to be flipped. For example, \[ r^{-3} \] Essentially means \[ \frac{1}{r^3} \] The negative sign indicates that we take the reciprocal of the base with the exponent turned positive. This rule can be extended to any negative exponent. For instance: \[ a^{-n} = \frac{1}{a^n} \] Remember to always convert negative exponents to their positive reciprocal form for simplification.
reciprocal of exponents
The reciprocal of an exponent involves flipping the base-and-exponent term. This is often necessary when working with negative exponents, as seen in our previous example. The reciprocal of \[ r^{-3} \] can be simplified to: \[ \frac{1}{r^3} \] This transformation is essential for simplification, especially in division problems. By converting negative exponents to reciprocals, you make the terms easier to handle and solve. For example, \[ a^{-m} \text{ becomes } \frac{1}{a^m} \] This makes complex expressions more straightforward and manageable.

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