Question

rewrite the following expression in simplest radical form given ( x > 0 ).
( sqrt4{256x^8} cdot sqrt{144x^5} )

Explanation:

Step1: Simplify the fourth root

We know that \(256 = 4^4\) and \(x^8=(x^2)^4\). Using the property \(\sqrt[n]{a^n}=a\) (for \(a\geq0\) and \(n\) even), we have:
\(\sqrt[4]{256x^8}=\sqrt[4]{4^4\cdot(x^2)^4}=4x^2\)

Step2: Simplify the square root

We know that \(144 = 12^2\) and \(x^5=x^4\cdot x\). Using the property \(\sqrt{a^2}=a\) (for \(a\geq0\)) and \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)), we have:
\(\sqrt{144x^5}=\sqrt{12^2\cdot x^4\cdot x}=\sqrt{12^2}\cdot\sqrt{x^4}\cdot\sqrt{x}=12x^2\sqrt{x}\)

Step3: Multiply the two simplified expressions

Now we multiply \(4x^2\) (from Step 1) and \(12x^2\sqrt{x}\) (from Step 2):
\(4x^2\cdot12x^2\sqrt{x}=(4\times12)\cdot(x^2\cdot x^2)\cdot\sqrt{x}\)
Using the property \(a^m\cdot a^n=a^{m + n}\), we get:
\(48x^{2+2}\sqrt{x}=48x^4\sqrt{x}\)

Answer:

\(48x^4\sqrt{x}\)