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Chapter 4: Problem 13

For Problems \(9-50\), simplify each rational expression. $$ \frac{-14 y^{3}}{56 x y^{2}} $$

Short Answer

Expert verified
The simplified expression is \(\frac{-y}{4x}\).

Step by step solution

01

Factorize the Numerator and Denominator

Factor the coefficients and variables in both the numerator and the denominator. The numerator is \(-14y^3\), which can be written as \(-1\times 14 \times y \times y \times y\). The denominator is \(56xy^2\), which can be written as \(56 \times x \times y \times y\).
02

Simplify the Coefficients

Simplify the coefficients \(-14\) in the numerator and \(56\) in the denominator by dividing both by their greatest common divisor, which is \(14\). This gives:\[\frac{-14}{56} = \frac{-1}{4}\].So the expression now becomes:\[\frac{-1 \times y \times y \times y}{4 \times x \times y \times y}\].
03

Simplify the Variables

Cancel common factors in the numerator and the denominator. We have common factors of \(y\) in both the numerator and denominator:\[\frac{y^3}{y^2} = y^{3-2} = y\].So the expression simplifies to:\[\frac{-1 \times y}{4 \times x}\].
04

Write the Simplified Expression

Combine the simplified terms from the previous steps to write the final simplified expression:\[\frac{-y}{4x}\].

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is a method used to simplify expressions by breaking them down into products of simpler factors. In our example, the expression is \(-14 y^3\). Here, we identify that \(-14\) is the coefficient and \(y^3\) can be expressed as \(y\times y\times y\).

Understanding how to factor polynomials involves recognizing that you can decompose variables into their base components along with multiplying integers (the coefficients).
  • The term \(14\) is already a product of its factors, which are \(1\) and \(14\).
  • For the variables, any term like \(y^3\) can be thought of as multiplying \(y\) three times (\(y \times y \times y\)).
Factoring plays a crucial role in simplifying any rational expression because it lets you see what common elements exist in both the numerator and the denominator.
Greatest Common Divisor
The greatest common divisor (GCD) is a useful tool for simplifying fractions by determining the largest number that evenly divides two or more numbers. In our rational expression \(-14/56\), we look at the GCD of \(-14\) and \(56\), which is \(14\).

Using the GCD helps you reduce fractions efficiently:
  • Divide both the numerator and the denominator by the GCD.
  • In this example, dividing both \(-14\) and \(56\) by \(14\) leaves us with \(-1/4\).
By simplifying the coefficients, we make it easier to see what remains to be simplified among the variables. This process is vital to making the rational expression as simplified as possible.
Variable Cancellation
Variable cancellation is an important step when simplifying expressions that involve variables in both the numerator and the denominator. In an expression like \(\frac{y^3}{y^2}\), observe that the \(y\)'s are common factors.

Here's how you cancel variables effectively:
  • Subtract the exponents of the common variable (\(y^3/y^2 = y^{3-2} = y\)).
  • Every common variable can be reduced, leading to simpler results.
Variable cancellation helps us eliminate unnecessary parts of an expression, making the resultant expression more manageable. Remember, only variables with the same base and present in both numerator and denominator can be cancelled.

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Most popular questions from this chapter

For Problems \(1-30\), solve each equation. $$ \frac{2}{n+3}+\frac{3}{n-4}=\frac{2 n-1}{n^{2}-n-12} $$

Felipe jogs for 10 miles and then walks another 10 miles. He jogs \(2 \frac{1}{2}\) miles per hour faster than he walks, and the entire distance of 20 miles takes 6 hours. Find the rate at which he walks and the rate at which he jogs.

For Problems \(1-30\), solve each equation. $$ \frac{-2}{3 x+2}+\frac{x-1}{9 x^{2}-4}=\frac{3}{12 x-8} $$

For Problems \(1-30\), solve each equation. $$ \frac{x+1}{2 x^{2}+7 x-4}-\frac{x}{2 x^{2}-7 x+3}=\frac{1}{x^{2}+x-12} $$

Why is it important to consider more than one way to do a problem?
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