Problem 79 - College Algebra

Question

Add or subtract terms whenever possible. $$\sqrt[3]{54 x y^{3}}-y \sqrt[3]{128 x}$$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt[3]{54} * \sqrt[3]{x} * y - 4y \sqrt[3]{2} \sqrt[3]{x}\)

Step 1: Simplify the First Term

The cube root of the product of multiple terms is equal to the product of the cube roots of these terms. Therefore, we can split \(\sqrt[3]{54 x y^{3}}\) into \(\sqrt[3]{54} * \sqrt[3]{x} * \sqrt[3]{y^{3}}\). The cube root of \(y^{3}\) simplifies to \(y\), so we have \(\sqrt[3]{54} * \sqrt[3]{x} * y\). The cube root of 54 cannot be simplified further, so it is left as it is

Step 2: Simplify the Second Term

Similarly to Step 1, split \(y \sqrt[3]{128 x}\) into \(y * \sqrt[3]{128} * \sqrt[3]{x}\). The square root of 128 can be simplified further by expressing 128 as the product of 64 and 2, leading to \(y * \sqrt[3]{64} * \sqrt[3]{2} * \sqrt[3]{x} = y * 4 * \sqrt[3]{2} * \sqrt[3]{x}\). Thus, the simplified form of \(y \sqrt[3]{128 x}\) is \(4y \sqrt[3]{2} \sqrt[3]{x}\)

Step 3: Express the Final Result

The original expression \(\sqrt[3]{54 x y^{3}} - y \sqrt[3]{128 x}\) simplifies to \(\sqrt[3]{54} * \sqrt[3]{x} * y - 4y \sqrt[3]{2} \sqrt[3]{x}\), which is the final result. These terms cannot be combined further, since the sub-terms \(\sqrt[3]{54}\) and \(4 \sqrt[3]{2}\) are not the same