Simplify by factoring. Assume the variable in the radicand is a positive real number: the cube root of 7 divided by y.

Step-by-step solution

1. 1

   Split the radical over the fraction. The quotient rule for radicals gives $\sqrt[3]{7/y} = \dfrac{\sqrt[3]{7}}{\sqrt[3]{y}}$. A cube root is still sitting in the denominator, so the expression is not yet in simplified form.

2. 2

   Decide what the denominator needs. To clear a cube root we need the radicand underneath to be a perfect cube. $y$ is one factor short of $y^3$ by two, so multiplying by $\sqrt[3]{y^2}$ will finish it.

3. 3

   Multiply by a form of 1. $\dfrac{\sqrt[3]{7}}{\sqrt[3]{y}} \cdot \dfrac{\sqrt[3]{y^2}}{\sqrt[3]{y^2}} = \dfrac{\sqrt[3]{7y^2}}{\sqrt[3]{y^3}}$.

4. 4

   Evaluate the denominator. Since $y > 0$, $\sqrt[3]{y^3} = y$, leaving $\dfrac{\sqrt[3]{7y^2}}{y}$.

5. 5

   Check numerically. At $y = 8$ the original is $\sqrt[3]{0.875} \approx 0.9565$, and $\sqrt[3]{7 \cdot 64}/8 = \sqrt[3]{448}/8 \approx 7.6517/8 \approx 0.9565$.