Simplify: $\frac{1}{3}(-18x + 36)$
The expression given is \(8 \cdot \frac{1}{3} \cdot (-18x + 36)\). Our goal is to simplify this expression by applying the distributive property of multiplication over addition. The constants outside the parentheses, \(8\) and \(\frac{1}{3}\), can be combined into a single factor \(\frac{8}{3}\), which we will then distribute to each term inside the parentheses, \(-18x\) and \(36\).
To simplify the expression, we first multiply the constants outside the parentheses:
\[8 \cdot \frac{1}{3} = \frac{8}{3}\]
Now, we distribute \(\frac{8}{3}\) to the terms inside the parentheses:
\[\frac{8}{3}(-18x + 36) = \left(\frac{8}{3} \cdot -18x\right) + \left(\frac{8}{3} \cdot 36\right)\]
Now we calculate the individual products:
First term: \(\frac{8}{3} \cdot -18x = 8 \cdot \frac{-18}{3}x = 8 \cdot (-6x) = -48x\)
Second term: \(\frac{8}{3} \cdot 36 = 8 \cdot \frac{36}{3} = 8 \cdot 12 = 96\)
Combining these, we get \(-48x + 96\).
The simplified expression is \(-48x + 96\). This can also be written as \(96 - 48x\). Both forms are equivalent. Thus, the simplified expression is \(\boxed{-48x + 96}\).
In real-life scenarios, simplifying expressions like this is common in finance or project management. For example, if a company's budget for a project starts with a base amount and needs to be adjusted based on a variable \(x\) representing the number of units produced, an expression like this helps quickly calculate the net adjustment to the budget regardless of the scale. Understanding how to distribute and combine terms allows for efficient planning and quick updates to numerical models.