Question

simplify.
\sqrt{54x^{9}}
assume that the variable represents a positive real number.

Explanation:

Step1: Factor the radicand

Factor \(54x^9\) into perfect - square factors and non - perfect - square factors.
We know that \(54 = 9\times6\) and \(x^9=x^{8}\times x=(x^{4})^{2}\times x\). So, \(\sqrt{54x^{9}}=\sqrt{9\times6\times x^{8}\times x}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a\geq0,b\geq0\))

According to the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we can split the square root:
\(\sqrt{9\times6\times x^{8}\times x}=\sqrt{9}\times\sqrt{6}\times\sqrt{x^{8}}\times\sqrt{x}\)

Step3: Simplify each square root

We know that \(\sqrt{9} = 3\), \(\sqrt{x^{8}}=x^{4}\) (since \((x^{4})^{2}=x^{8}\) and \(x\) is a positive real number).
So, \(\sqrt{9}\times\sqrt{6}\times\sqrt{x^{8}}\times\sqrt{x}=3\times x^{4}\times\sqrt{6x}\) (because \(\sqrt{6}\times\sqrt{x}=\sqrt{6x}\))

Answer:

\(3x^{4}\sqrt{6x}\)