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Chapter 7: Problem 31

In the following exercises, multiply the rational expressions. $$ \frac{5 x^{2} y^{4}}{12 x y^{3}} \cdot \frac{6 x^{2}}{20 y^{2}} $$

Short Answer

Expert verified
\(\frac{x^{3}}{8 y}\)

Step by step solution

01

- Multiply the Numerators

First, multiply the numerators of the given fractions: \[5 x^{2} y^{4} \times 6 x^{2} = 30 x^{4} y^{4}\]
02

- Multiply the Denominators

Next, multiply the denominators of the given fractions: \[12 x y^{3} \times 20 y^{2} = 240 x y^{5}\]
03

- Combine the Fractions

Combine the results from Step 1 and Step 2 into a single rational expression: \[\frac{30 x^{4} y^{4}}{240 x y^{5}}\]
04

- Simplify the Coefficients

Simplify the coefficients (numerical parts) of the fraction: \[\frac{30}{240} = \frac{1}{8}\]
05

- Simplify the Variables

Simplify the variables in both the numerator and the denominator: \[\frac{x^{4}}{x} \cdot \frac{y^{4}}{y^{5}} = x^{4-1} \cdot y^{4-5} = x^{3} y^{-1}\]
06

- Finalize the Expression

Combine the simplified coefficients and variables together: \[\frac{1}{8} x^{3} y^{-1}\]This simplifies further to: \[\frac{x^{3}}{8 y}\]

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Key Concepts

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

rational expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Polynomials can include variables, coefficients, and constants combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding rational expressions is important because they show up frequently in algebra, and knowing how to manipulate them will help solve more complex algebraic equations.
Example of a rational expression: \( \frac{3x^2 + 2x - 5}{x^2 - 1} \)
simplifying variables
Simplifying variables means reducing the expression to its simplest form by canceling out the common factors in the numerator and the denominator. When simplifying variables, follow these steps:
  • Identify common variables in both the numerator and the denominator.
  • Subtract the exponent in the denominator from the exponent in the numerator for each variable.
  • If the result is a positive exponent, the variable remains in the numerator; if negative, it remains in the denominator.
Example from our exercise: \ \( \frac{x^4}{x} = x^{4-1} = x^3 \) and \ \( \frac{y^4}{y^5} = y^{4-5} = y^{-1} = \frac{1}{y} \).
So, \ \frac{x^3}{8y} \ is the simplified form.
multiplying fractions
Multiplying fractions is straightforward. You multiply the numerators together to get the new numerator and the denominators together to get the new denominator.
  • Multiply the numerators: \( 5x^2y^4 \times 6x^2 = 30x^4y^4 \).
  • Multiply the denominators: \( 12xy^3 \times 20y^2 = 240xy^5 \).
  • Combine them into one fraction: \( \frac{30x^4y^4}{240xy^5} \).
This step is essential before simplifying the expression further.
algebraic fractions
Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. They function similarly to regular fractions, but with the added step of dealing with variables.
Here's how you handle them:
  • Perform operations (addition, subtraction, multiplication, or division) as you would with numerical fractions.
  • Ensure the expressions are in their simplest form by factoring where possible.
  • Remember the basic rules of exponents when multiplying or dividing terms.
From our example, after multiplying and simplifying, we kept the rational expression balanced by treating the variables just as carefully as we did the numerical parts. Final expression: \( \frac{x^{3}}{8y} \)

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Most popular questions from this chapter

Joseph is traveling on a road trip. The distance, \(d,\) he travels before stopping for lunch varies directly with the speed, \(v,\) he travels. He can travel 120 miles at a speed of 60 mph. (a) Write the equation that relates \(d\) and \(v\). (b) How far would he travel before stopping for lunch at a rate of 65 mph?

Solve the application problem provided. Darrin can skateboard 2 miles against a 4 -mph wind in the same amount of time he skateboards 6 miles with a 4 -mph wind. Find the speed Darrin skateboards with no wind.

If \(a\) varies directly as \(b\) and \(a=16,\) when \(b=4\). find the equation that relates \(a\) and \(b\).

Solve. $$m=\frac{n}{2-n} \text { for } n .$$

Solve each rational equation. $$\frac{2}{x^{2}+2 x-3}+\frac{3}{x^{2}+4 x+3}=\frac{6}{x^{2}-1}$$
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