Add or subtract terms whenever possible: the cube root of 16xy³ minus y times the cube root of 128x.

Step-by-step solution

1. 1

   Understand the obstacle. Radicals can only be combined when their index and radicand match exactly. Here the radicands $16xy^3$ and $128x$ differ, so each must be simplified before anything can be subtracted.

2. 2

   Simplify the first radical. $16xy^3 = 8y^3 \cdot 2x$, and both $8$ and $y^3$ are perfect cubes, so $\sqrt[3]{16xy^3} = 2y\sqrt[3]{2x}$.

3. 3

   Simplify the second radical. $128x = 64 \cdot 2x$ and $64 = 4^3$, so $\sqrt[3]{128x} = 4\sqrt[3]{2x}$ and the whole term is $4y\sqrt[3]{2x}$.

4. 4

   Now the radicals match. Both terms carry the common radical $\sqrt[3]{2x}$, so they behave like $2y$ and $4y$ copies of the same object.

5. 5

   Subtract the coefficients. $2y\sqrt[3]{2x} - 4y\sqrt[3]{2x} = -2y\sqrt[3]{2x}$.

6. 6

   Check numerically. At $x = 1$, $y = 1$: $\sqrt[3]{16} - \sqrt[3]{128} \approx 2.5198 - 5.0397 = -2.5198$, and $-2\sqrt[3]{2} \approx -2.5198$.