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Chapter 0: Problem 79

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

Short Answer

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

Step by step solution

01

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
02

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 \(4y \sqrt[3]{2} \sqrt[3]{x}\)
03

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

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Most popular questions from this chapter

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