Questions: Multiply as indicated. If possible, simplify any radical expressions that appear in the product. Assume that all variables represent positive real numbers. ∛x(∛(250 x^2) - ∛x)

Solution Steps

To solve this problem, we need to distribute the cube root of \(x\) across the terms inside the parentheses. This involves multiplying the cube roots and simplifying the resulting expressions. We will use the property that \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\).

We start with the expression: \[ \sqrt[3]{x}\left(\sqrt[3]{250 x^{2}} - \sqrt[3]{x}\right) \] Using the property of cube roots, we distribute \(\sqrt[3]{x}\) across the terms inside the parentheses: \[ \sqrt[3]{x} \cdot \sqrt[3]{250 x^{2}} - \sqrt[3]{x} \cdot \sqrt[3]{x} \]

The first term simplifies as follows: \[ \sqrt[3]{x} \cdot \sqrt[3]{250 x^{2}} = \sqrt[3]{250 x^{3}} = 5 \cdot \sqrt[3]{2} \cdot x \] The second term simplifies to: \[ \sqrt[3]{x} \cdot \sqrt[3]{x} = \sqrt[3]{x^2} \] Thus, the expression becomes: \[ 5 \cdot \sqrt[3]{2} \cdot x - \sqrt[3]{x^2} \]

Combining the terms, we have: \[

• \sqrt[3]{x^2} + 5 \cdot \sqrt[3]{2} \cdot x \]

Final Answer