Question

simplify the expression (8sqrt3{320x^{9}})

Explanation:

Step1: Factor the radicand

First, factor 320 and \(x^9\) into cubes and other factors. We know that \(320 = 64\times5=4^3\times5\) and \(x^9=(x^3)^3\). So the expression inside the cube root, \(320x^9\), can be written as \(4^3\times5\times(x^3)^3\).
The original expression \(8\sqrt[3]{320x^9}\) becomes \(8\sqrt[3]{4^3\times5\times(x^3)^3}\).

Step2: Simplify the cube root

Using the property of cube roots \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) (where \(a = 4^3\times(x^3)^3\) and \(b = 5\)), we can separate the cube - root terms:
\(8\sqrt[3]{4^3\times(x^3)^3\times5}=8\times\sqrt[3]{4^3}\times\sqrt[3]{(x^3)^3}\times\sqrt[3]{5}\)
Since \(\sqrt[3]{4^3} = 4\) and \(\sqrt[3]{(x^3)^3}=x^3\), we substitute these values back into the expression:
\(8\times4\times x^3\times\sqrt[3]{5}\)

Step3: Multiply the constant terms

Multiply 8 and 4: \(8\times4 = 32\). So the simplified expression is \(32x^3\sqrt[3]{5}\)

Answer:

\(32x^3\sqrt[3]{5}\)