- Factor each radical term to extract perfect squares: \(\sqrt{20uv^2} = 2v\sqrt{5u}\) and \(\sqrt{45u^3} = 3u\sqrt{5u}\).
- Multiply by the respective leading coefficients to get \(16uv\sqrt{5u}\) and \(3uv\sqrt{5u}\).
- Combine the like terms by adding the coefficients: \(16uv\sqrt{5u} + 3uv\sqrt{5u} = 19uv\sqrt{5u}\).
- Final simplified result is \(\boxed{19uv\sqrt{5u}}\).
Explanation
- Analysis of the Expression
The given expression is \(8u\sqrt{20uv^2} + v\sqrt{45u^3}\), where \(u\) and \(v\) are positive real numbers. To simplify this, we need to extract perfect squares from inside the radicals. We recall the property of radicals \(\sqrt{a^2b} = a\sqrt{b}\) for \(a \ge 0\). Given that \(u, v > 0\), we can safely use this property.
- Simplifying the First Term
First, let us simplify the first term: \(8u\sqrt{20uv^2}\).
We can factor the constant and variables inside the square root to identify perfect squares:
\(\sqrt{20uv^2} = \sqrt{(4 \times 5) \times u \times v^2} = \sqrt{4v^2 \times 5u} = \sqrt{(2v)^2 \times 5u} = 2v\sqrt{5u}\).
Now, multiply this by the coefficient \(8u\):
\(8u \times 2v\sqrt{5u} = 16uv\sqrt{5u}\).
- Simplifying the Second Term
Next, let us simplify the second term: \(v\sqrt{45u^3}\).
Similarly, factor the expression inside the square root:
\(\sqrt{45u^3} = \sqrt{(9 \times 5) \times u^2 \times u} = \sqrt{9u^2 \times 5u} = \sqrt{(3u)^2 \times 5u} = 3u\sqrt{5u}\).
Now, multiply this by the coefficient \(v\):
\(v \times 3u\sqrt{5u} = 3uv\sqrt{5u}\).
- Combining the Terms and Finalizing
Now that both terms have been simplified to the form \(k \cdot uv\sqrt{5u}\), we can combine them:
\(16uv\sqrt{5u} + 3uv\sqrt{5u} = (16 + 3)uv\sqrt{5u} = 19uv\sqrt{5u}\).
Thus, the expression simplifies to \(19uv\sqrt{5u}\).
Examples
In engineering, simplifying algebraic expressions containing radicals is essential for calculating physical quantities such as forces, velocities, or resistances. For instance, when calculating the magnitude of a resulting vector in physics, you often end up with combined radical terms. Simplification, as demonstrated here, allows for a more efficient and cleaner analytical model, which simplifies further computations, simulation, or design specifications. By factoring out common factors, you reduce complexity and minimize the possibility of errors in subsequent calculations.