- Simplify the first term: \(9 u \sqrt{3 u w^2} = 9 u w \sqrt{3 u}\).
- Simplify the second term: \(w \sqrt{48 u^3} = 4 u w \sqrt{3 u}\).
- Subtract the second simplified term from the first: \(9 u w \sqrt{3 u} - 4 u w \sqrt{3 u} = 5 u w \sqrt{3 u}\).
- The final simplified expression is \(\boxed{5 u w \sqrt{3 u}}\).
Explanation
- Problem Analysis
We are given the expression \(9 u \sqrt{3 u w^2} - w \sqrt{48 u^3}\) and are told that \(u, w > 0\). The goal is to simplify this expression as much as possible by extracting perfect squares from underneath the square root radicals and combining like terms. Because both \(u\) and \(w\) are positive, we can safely use the property \(\sqrt{x^2} = x\) for any \(x > 0\).
- Simplifying the First Term
First, we simplify the term \(9 u \sqrt{3 u w^2}\). We recognize that \(w^2\) is a perfect square inside the radical. Using the properties of square roots, specifically \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we can rewrite the radical as follows:
\[\sqrt{3 u w^2} = \sqrt{w^2} \cdot \sqrt{3 u} = w \sqrt{3 u}.\]
Substituting this back into the original term, we get:
\[9 u (w \sqrt{3 u}) = 9 u w \sqrt{3 u}.\]
- Simplifying the Second Term
Next, we simplify the second term, \(w \sqrt{48 u^3}\). First, we factor the radicand \(48 u^3\) to identify perfect squares:
\[48 u^3 = 16 \cdot 3 \cdot u^2 \cdot u.\]
Taking the square root of the perfect square factors:
\[\sqrt{48 u^3} = \sqrt{16 \cdot u^2 \cdot 3 \cdot u} = \sqrt{16} \cdot \sqrt{u^2} \cdot \sqrt{3 u} = 4 \cdot u \cdot \sqrt{3 u} = 4 u \sqrt{3 u}.\]
Now, multiplying this by the \(w\) outside the radical, we obtain:
\[w (4 u \sqrt{3 u}) = 4 u w \sqrt{3 u}.\]
- Final Simplification
Now that both terms have been simplified to involve the same radical \(\sqrt{3 u}\) and the same variables \(u\) and \(w\), we can subtract them:
\[9 u w \sqrt{3 u} - 4 u w \sqrt{3 u}.\]
Since both terms are like terms, we can factor out the common part \(u w \sqrt{3 u}\):
\[(9 - 4) u w \sqrt{3 u} = 5 u w \sqrt{3 u}.\]
Thus, the simplified expression is \(\boxed{5 u w \sqrt{3 u}}\).
Examples
Imagine you are an architect calculating the cost of materials for a project where the layout involves specific geometric patterns. Suppose you have an area defined by two different structural components whose dimensions follow the expression \(9 u \sqrt{3 u w^2} - w \sqrt{48 u^3}\). By simplifying this to \(5 u w \sqrt{3 u}\), you can quickly estimate the total material required based on variables \(u\) and \(w\) without having to perform tedious calculations for each part individually. This algebraic simplification saves time and reduces the likelihood of errors in complex structural estimations.