Simplify. Express your answer as a single fraction in simplest form.
$\frac{w^4 x}{3}-(x+2)$
We are given the expression \(\frac{w^4 x}{3} - (x + 2)\). Our goal is to simplify this expression and write it as a single fraction in its simplest form. The expression involves a term with a fraction and a binomial being subtracted. To combine these, we need to find a common denominator.
First, we distribute the negative sign to the terms inside the parentheses. The expression \(-(x+2)\) becomes \(-x - 2\). Now, our expression is:
\[\frac{w^4 x}{3} - x - 2\]
To combine all parts into a single fraction, we need a common denominator of 3 for all terms.
The term \(-x\) can be written as \(\frac{3 \cdot x}{3} = \frac{3x}{3}\).
The constant \(-2\) can be written as \(\frac{3 \cdot 2}{3} = \frac{6}{3}\).
Substituting these back into our expression, we have:
\[\frac{w^4 x}{3} - \frac{3x}{3} - \frac{6}{3}\]
Now that all terms have the same denominator, we can combine the numerators over this common denominator:
\[\frac{w^4 x - 3x - 6}{3}\]
This is the single fraction in its simplest form.
Simplifying algebraic expressions is a fundamental skill in algebra, useful for solving equations, working with functions, or modeling real-world scenarios. For example, if you are modeling the cost of a production process where \(w\) represents the number of batches and \(x\) represents the unit cost, and you need to subtract fixed costs or additional variable costs from a total revenue fraction, you would use this skill to simplify the resulting expression for easier analysis. This simplifies the management of complex equations in fields like economics or physics.