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Chapter 4: Problem 96

Simplify the expression. $$-\frac{3 m^{2} n}{9 m^{2} n^{4}}$$

Short Answer

Expert verified
-\frac{1}{3 n^3}

Step by step solution

01

- Identify like terms

In the expression \(-\frac{3 m^{2} n}{9 m^{2} n^{4}}\), we need to simplify the terms appearing in both the numerator and the denominator. Here, the like terms are \(m^2\) and \(n\).
02

- Simplify the numerical coefficients

The coefficients in the expression are -3 (numerator) and 9 (denominator). Simplify this fraction: \(-\frac{3}{9} = -\frac{1}{3}\).
03

- Simplify the variable \(m\)

In both the numerator and the denominator, \(m^2\) appears. Therefore, they cancel each other out: \(m^2/m^2 = 1\).
04

- Simplify the variable \(n\)

The variable \(n\) in the numerator cancels with one \(n\) in the denominator, leaving \(n^3\) in the denominator. So, \(-\frac{3 m^{2} n}{9 m^{2} n^{4}} = -\frac{1}{3 n^3}\).

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Key Concepts

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

Algebraic Simplification
Algebraic simplification is all about making expressions easier to work with. In algebra, expressions can get complex. Our goal is to break them down into their simplest form.
For instance, if you encounter \(-\frac{3 m^{2} n}{9 m^{2} n^{4}}\), you should look for terms that can be reduced.
This means identifying common factors in both the numerator and denominator, and then reducing the fraction.
Simplification helps not only in solving equations faster but also in better understanding the relationships between variables.
Rational Expressions
A rational expression is a fraction where the numerator and denominator are polynomials.
Just like numerical fractions, rational expressions can and should be simplified when possible.
Let's use the example from the exercise: \(-\frac{3 m^{2} n}{9 m^{2} n^{4}}\).
Here are the steps to simplify:
  • Find and factor out common terms.
  • Simplify numerical coefficients.
  • Reduce the variables.
By treating the numerator and denominator separately, you can break them down into more manageable parts.
Reducing Fractions
Reducing fractions means making them as simple as they can be. This involves dividing the top and bottom by their greatest common divisor (GCD).
With algebraic fractions, this also involves canceling out common variables.
For instance, in \(-\frac{3 m^{2} n}{9 m^{2} n^{4}}\), we do the following:
  • Simplify \(-\frac{3}{9}\) to \(-\frac{1}{3}\)
  • Cancel \m^{2}\ in both numerator and denominator
  • Simplify the variable \ such that we are left with \^3\ in the denominator
This results in \(-\frac{1}{3 n^3}\), which is our simplified fraction.

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Most popular questions from this chapter

Simplify the fraction to lowest terms. Write the answer as a fraction or a whole number. $$\frac{2}{2}$$

Convert the improper fraction to a mixed number. $$-\frac{23}{6}$$

Fill in the blanks with \(<,>,\) or \(=\). Which of the following fractions has the greatest value? \(? \frac{2}{3}, \frac{7}{8}, \frac{5}{6}, \frac{1}{2}\)

Find the LCM. 42 and 70

One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out every sccond number after the number 2. This will climinate all numbers that are multiples of 2. Then go back to the beginning of the chart and leave the number 3 , but cross out every third number after the number 3 (thus eliminating the multiples of 3 ). Begin at the next open number and continue this process. The numbers that remain are prime numbers. Use this process to find the prime numbers less than 50 . $$\begin{array}{c|c|c|c|c|c|c|c|c|c} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ \hline \end{array}$$
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