- Simplify the leading radical term: \(2\sqrt{8x^3} = 4x\sqrt{2x}\).
- Simplify the terms inside the parentheses: \(3\sqrt{10x^4} - x\sqrt{5x^2} = 3x^2\sqrt{10} - x^2\sqrt{5}\).
- Expand the expression by multiplying the simplified terms: \(4x\sqrt{2x} \cdot (3x^2\sqrt{10} - x^2\sqrt{5}) = 24x^3\sqrt{5x} - 4x^3\sqrt{10x}\).
- Final simplified product: \(\boxed{24x^3 \sqrt{5x} - 4x^3 \sqrt{10x}}\).
Explanation
- Simplifying the leading term
To simplify the expression \(2 \sqrt{8 x^3} (3 \sqrt{10 x^4} - x \sqrt{5 x^2})\) with \(x \geq 0\), we start by simplifying the radical terms individually.
First, consider \(2 \sqrt{8 x^3}\):
\[\sqrt{8 x^3} = \sqrt{4 \cdot 2 \cdot x^2 \cdot x} = 2x \sqrt{2x}\]
Multiplying by the leading 2, we have:
\[2 \sqrt{8 x^3} = 2 \cdot (2x \sqrt{2x}) = 4x \sqrt{2x}\]
- Simplifying terms inside the parentheses
Next, simplify the terms inside the parentheses:
\[3 \sqrt{10 x^4} = 3 \cdot x^2 \sqrt{10} = 3x^2 \sqrt{10}\]
\[x \sqrt{5 x^2} = x \cdot x \sqrt{5} = x^2 \sqrt{5}\]
So the expression becomes:
\[4x \sqrt{2x} \cdot (3x^2 \sqrt{10} - x^2 \sqrt{5})\]
- Expanding and finalizing the simplification
Distribute \(4x \sqrt{2x}\) across the parentheses:
\[(4x \sqrt{2x} \cdot 3x^2 \sqrt{10}) - (4x \sqrt{2x} \cdot x^2 \sqrt{5})\]
For the first part:
\[12x^3 \sqrt{2x \cdot 10} = 12x^3 \sqrt{20x} = 12x^3 \sqrt{4 \cdot 5x} = 12x^3 \cdot 2 \sqrt{5x} = 24x^3 \sqrt{5x}\]
For the second part:
\[4x^3 \sqrt{2x \cdot 5} = 4x^3 \sqrt{10x}\]
Combining these, we get:
\[24x^3 \sqrt{5x} - 4x^3 \sqrt{10x}\]
- Conclusion
Comparing our result \(24x^3 \sqrt{5x} - 4x^3 \sqrt{10x}\) to the provided options, it matches the second option. The final answer is \(\boxed{24x^3 \sqrt{5x} - 4x^3 \sqrt{10x}}\).
Examples
Simplifying expressions with radicals is essential in physics and engineering, especially when dealing with formulas involving square roots, such as calculating kinetic energy, electrical impedance, or distances. For instance, in structural engineering, when calculating the stress on beams of various dimensions represented by algebraic expressions, simplifying these expressions allows for easier determination of material requirements. Accurate algebraic manipulation ensures that designs are safe, efficient, and cost-effective.