Questions: Rationalize the denominator and simplify if possible. (sqrt[3]4)/(sqrt[3]9 z^2)

Solution Steps

To rationalize the denominator, we need to eliminate the cube root in the denominator. We can do this by multiplying both the numerator and the denominator by the cube root of an appropriate expression that will make the denominator a perfect cube.

1. Identify the cube root in the denominator.
2. Multiply both the numerator and the denominator by the cube root of a value that will make the denominator a perfect cube.
3. Simplify the resulting expression.

We start with the expression

\[ \frac{\sqrt[3]{4}}{\sqrt[3]{9 z^{2}}} \]

To eliminate the cube root in the denominator, we multiply both the numerator and the denominator by

\[ \sqrt[3]{3^2 z^{2}} = 3^{2/3} z^{2/3} \]

This gives us:

\[ \frac{\sqrt[3]{4} \cdot \sqrt[3]{3^2 z^{2}}}{\sqrt[3]{9 z^{2}} \cdot \sqrt[3]{3^2 z^{2}}} \]

The new numerator becomes:

\[ \sqrt[3]{4} \cdot \sqrt[3]{9} \cdot z^{2/3} = \sqrt[3]{36} \cdot z^{2/3} \]

The new denominator simplifies to:

\[ \sqrt[3]{(9 z^{2}) \cdot (9 z^{2})} = \sqrt[3]{81 z^{4}} = 3 \cdot \sqrt[3]{9 z^{4}} \]

Thus, we have:

\[ \frac{\sqrt[3]{36} \cdot z^{2/3}}{3 \cdot \sqrt[3]{9 z^{4}}} \]

Final Answer