Problem 96 Simplify the expression. $$-\f... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

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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

- 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 $n$.

- Simplify the numerical coefficients

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

- Simplify the variable $m$

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

- 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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91影视

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

Short Answer

Step by step solution

- Identify like terms

- Simplify the numerical coefficients

- Simplify the variable \(m\)

- Simplify the variable \(n\)

Key Concepts

Algebraic Simplification

Rational Expressions

Reducing Fractions

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