Question

simplify.
\\(\sqrt{24x^{10}}\\)
assume that the variable \\(x\\) represents a positive real number.

Explanation:

Step1: Factor the radicand

We can factor \(24x^{10}\) as \(4\times6\times x^{10}\). Since \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)), we can split the square root: \(\sqrt{24x^{10}}=\sqrt{4\times6\times x^{10}}\).

Step2: Simplify each square root

We know that \(\sqrt{4} = 2\) and for \(x\) positive, \(\sqrt{x^{10}}=x^{5}\) (because \((x^{5})^{2}=x^{10}\)). Then we have \(\sqrt{4\times6\times x^{10}}=\sqrt{4}\cdot\sqrt{6}\cdot\sqrt{x^{10}} = 2\cdot\sqrt{6}\cdot x^{5}\).

Step3: Combine the terms

Combining the simplified terms, we get \(2x^{5}\sqrt{6}\).

Answer:

\(2x^{5}\sqrt{6}\)