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Chapter 6: Problem 18

Find the reciprocal of each rational expression. \(\frac{6 x^{4}}{9 y^{2}}\)

Short Answer

Expert verified
The reciprocal of \(\frac{6 x^{4}}{9 y^{2}}\) is \(\frac{3 y^{2}}{2 x^{4}}\).

Step by step solution

01

Understand the Concept of a Reciprocal

The reciprocal of a number is 1 divided by that number. For any rational expression, its reciprocal will be the flipping of the numerator and the denominator.
02

Identify the Given Rational Expression

The rational expression given is \(\frac{6 x^{4}}{9 y^{2}}\).
03

Flip the Numerator and Denominator

To find the reciprocal, exchange the numerator and the denominator. Therefore, the reciprocal of \(\frac{6 x^{4}}{9 y^{2}}\) is \(\frac{9 y^{2}}{6 x^{4}}\).
04

Simplify the Reciprocal if Possible

Check if the reciprocal can be simplified further. \(\frac{9 y^{2}}{6 x^{4}}\) simplifies to \(\frac{3 y^{2}}{2 x^{4}}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

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Key Concepts

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

rational expressions
Rational expressions are fractions involving polynomials. They represent ratios of two polynomials. These expressions can often be simplified by cancelling common factors in the numerator and the denominator

For example, consider the expression \(\frac{6 x^{4}}{9 y^{2}}\). Here, \('6 x^{4}'\) is the numerator and \('9 y^{2}'\) is the denominator. Understanding how to manipulate and simplify these expressions forms a core skill in algebra.

Rational expressions can sometimes be intimidating, but with the right methods, you can tackle them confidently. Let's break down the main steps.
  • Identify the numerator and the denominator.
  • Look for common factors in both parts.
  • Simplify by reducing these common factors.
simplifying fractions
Simplifying fractions is all about breaking them down to their simplest form. This often involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both parts by this number.

For instance, in the original problem, after flipping \(\frac{6 x^{4}}{9 y^{2}}\) to find the reciprocal as \(\frac{9 y^{2}}{6 x^{4}}\), you need to simplify it.

The greatest common divisor of 9 and 6 is 3. Therefore, you divide both the numerator and the denominator by 3:
  • Numerator: \(\frac{9 y^{2}}{3} = 3 y^{2}\)
  • Denominator: \(\frac{6 x^{4}}{3} = 2 x^{4}\)
So, \(\frac{9 y^{2}}{6 x^{4}}\) simplifies to \(\frac{3 y^{2}}{2 x^{4}}\).

Simplifying fractions makes them easier to work with in equations and further calculations.
basic algebra
Basic algebra is the foundation of many mathematical concepts and is essential to solving rational expressions and other equations. The key principles you need to understand include:

  • Variables: Symbols that represent unknown values.
  • Exponents: Indicate how many times a number is multiplied by itself. For example, \('x^{4}'\) means \('x'\) is multiplied by itself four times.
  • Fractions: Represent division and can often be simplified by finding common factors.


In the example problem, we see how these basic concepts help us find and simplify the reciprocal of a rational expression.

Start with the given rational expression \(\frac{6 x^{4}}{9 y^{2}}\). Understanding the role of numerators, denominators, and the relationships between them is crucial. By flipping the expression to find its reciprocal and simplifying it, you utilize several fundamental algebraic techniques.

Mastering these basics will make more complex problems more approachable.

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Most popular questions from this chapter

Solve each problem. The average number of vehicles waiting in line to enter a sports arena parking area is approximated by the rational expression $$\frac{x^{2}}{2(1-x)}$$ where \(x\) is a quantity between 0 and 1 known as the traffic intensity. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Control, John Wiley and Sons. To the nearest tenth, find the average number of vehicles waiting if the traffic intensity is the given number. (a) 0.1 (b) 0.8 (c) 0.9 (d) What happens to the number of vehicles waiting as traffic intensity increases?

Simplify each expression, using only positive exponents in the answer. $$ \frac{k^{-1}+p^{-2}}{k^{-1}-3 p^{-2}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{m^{3} p}+\frac{2}{m p^{2}}}{\frac{4}{m p}+\frac{1}{m^{2} p}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{x^{3}-y^{3}}}{\frac{1}{x^{2}-y^{2}}} $$

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$
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