To rationalize the numerator $\frac{\sqrt[5]{x^4 z}}{y}$, we aim to eliminate the fifth root from the numerator.

The expression in the numerator is $\sqrt[5]{x^4 z}$, which is the fifth root of $x^4 z$. To rationalize it, we multiply both the numerator and denominator by an appropriate term so that the root becomes a whole power.

Here’s the step-by-step process:

Step 1: Identify the needed factor

We currently have $\sqrt[5]{x^4 z}$ in the numerator. To eliminate the fifth root, we need to multiply by a term that will make the power of $x^4$ and $z$ inside the radical a multiple of 5.

• For $x^4$, we need to multiply by $\sqrt[5]{x}$ to make the exponent 5.
• For $z$, we need to multiply by $\sqrt[5]{z^4}$ to make the exponent 5.

Thus, we multiply both the numerator and denominator by $\sqrt[5]{x z^4}$.

Step 2: Multiply the numerator and denominator

Multiply both the numerator and denominator by $\sqrt[5]{x z^4}$:

[ \frac{\sqrt[5]{x^4 z}}{y} \times \frac{\sqrt[5]{x z^4}}{\sqrt[5]{x z^4}} = \frac{\sqrt[5]{x^4 z} \times \sqrt[5]{x z^4}}{y \times \sqrt[5]{x z^4}} ]

Step 3: Simplify the numerator

Using the property of radicals $\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \cdot b}$, we get:

[ \sqrt[5]{(x^4 z) \cdot (x z^4)} = \sqrt[5]{x^5 z^5} = x z ]

So the numerator becomes $xz$, and the denominator is $y \cdot \sqrt[5]{x z^4}$.

Final Result:

The rationalized expression is:

$\frac{xz}{y \cdot \sqrt[5]{x z^4}}$

Would you like more details or have any other questions?

Here are 5 related questions:

1. How do you rationalize a denominator containing radicals?
2. What is the general approach for rationalizing cube roots?
3. How does rationalizing higher-order roots differ from square roots?
4. Can you rationalize an expression involving multiple variables inside a radical?
5. How do fractional exponents relate to radicals?

Tip:

Always check if the indices of radicals or exponents can be combined or simplified before multiplying for rationalization!