Simplify the radical: $\sqrt{60 x^{10}}$
To simplify the radical \(\sqrt{60 x^{10}}\), we begin by breaking down the expression into its constant and variable parts: \(\sqrt{60 x^{10}} = \sqrt{60} \cdot \sqrt{x^{10}}.\) This approach allows us to simplify each component individually before combining them.
Next, we simplify the constant part, \(\sqrt{60}\). We look for the largest perfect square that divides 60. Since \(60 = 4 \times 15\), we can write:
\[\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \cdot \sqrt{15} = 2 \sqrt{15}.\]
This provides the numerical part of our simplified expression.
Now, we simplify the variable part, \(\sqrt{x^{10}}\). Using the properties of exponents, specifically \(\sqrt{x^n} = x^{n/2}\) for \(x \ge 0\), we have:
\[\sqrt{x^{10}} = (x^{10})^{1/2} = x^{10/2} = x^5.\]
Combining the constant and variable parts, we obtain the full expression:
\[2 \sqrt{15} \cdot x^5 = 2 x^5 \sqrt{15}.\]
This matches option (B).
Simplifying radicals is a fundamental skill used in various fields, such as engineering and physics, to manage complex formulas and make them easier to work with. For instance, in structural engineering, when calculating the stress on a beam, formulas often contain square roots of physical dimensions. By simplifying these expressions, engineers can quickly assess if the design meets safety standards without needing to compute complex decimal values at every stage of the calculation. This process improves both efficiency and accuracy in real-world problem-solving.