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Chapter 6: Problem 17

Find the reciprocal of each rational expression. \(\frac{3 p^{3}}{16 q}\)

Short Answer

Expert verified
The reciprocal is \(\frac{16 q}{3 p^{3}}\).

Step by step solution

01

- Understand the Reciprocal

Understanding the reciprocal of a rational expression involves flipping the numerator and the denominator. This means that if you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
02

- Identify the Numerator and Denominator

In the given expression \(\frac{3 p^{3}}{16 q}\), the numerator is \(\text{3p}^3\) and the denominator is \(\text{16q}\).
03

- Flip the Numerator and Denominator

To find the reciprocal, switch the numerator and denominator. The reciprocal of \(\frac{3 p^{3}}{16 q}\) is \(\frac{16 q}{3 p^{3}}\).

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Key Concepts

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

Understanding Rational Expressions
Rational expressions are like fractions but with polynomials. When we talk about rational expressions, we are dealing with an expression that can be written as the quotient of two polynomials.

For example, in \(\frac{3 p^{3}}{16 q}\), we have a polynomial on the top and another on the bottom. Such expressions are called rational expressions because they can be expressed as the ratio of two polynomial expressions.

It's important to understand rational expressions because they appear in various areas of algebra, calculus, and even in real-life applications.
  • They can simplify complex problems.
  • They allow us to understand relations and functions better.
  • They're essential in solving equations involving polynomials.
Identifying the Numerator and Denominator
In any fraction, including rational expressions, the top part is the numerator, and the bottom part is the denominator.

In the expression \(\frac{3 p^{3}}{16 q}\), we see that:
  • The numerator is \(\text{3p}^3 \).
  • The denominator is \(\text{16q} \).


Understanding which is the numerator and which is the denominator is crucial because when you perform operations like finding the reciprocal, you need to switch these roles. The reciprocal involves flipping the numerator and denominator. This makes it vital to correctly identify these parts in the first place.
Finding the Reciprocal
To find the reciprocal of a rational expression, you flip the numerator and the denominator.

Essentially, if your rational expression is \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

Let's look at our exercise, \(\frac{3 p^{3}}{16 q}\). The steps to find its reciprocal are:
  • Identify the numerator and the denominator (which we already did).
  • Flip the numerator and denominator.
So, the reciprocal of \(\frac{3 p^{3}}{16 q}\) is \(\frac{16 q}{3 p^{3}}\). Remember that finding the reciprocal is straightforward once you know how to identify and switch the numerator and the denominator.

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Most popular questions from this chapter

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ \frac{2 q}{7}-\frac{q}{6+\frac{8}{4+4}} $$

Solve each formula or equation for the specified variable. $$ \frac{2}{r}+\frac{3}{s}+\frac{1}{t}=1 \text { for } t $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{2}{x-1}+2}{\frac{2}{x-1}-2} $$

Solve each formula or equation for the specified variable. $$ -3 t-\frac{4}{p}=\frac{6}{s} \text { for } p $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{3}{m}-m}{\frac{3-m^{2}}{4}} $$
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