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Radical form into exponential form  and exponential form into radical form

The given expression is:

\[
\sqrt[4]{(3xyz)^3}
\]

### Converting from Radical Form to Exponential Form:

In general, the expression \( \sqrt[n]{a^m} \) can be rewritten in exponential form as \( a^{m/n} \).

For the provided expression:

\[
\sqrt[4]{(3xyz)^3} = (3xyz)^{\frac{3}{4}}
\]

Thus, the exponential form of the expression is:

\[
(3xyz)^{\frac{3}{4}}
\]

### Converting from Exponential Form to Radical Form:

To convert an exponential expression like \( a^{m/n} \) back to radical form, it becomes \( \sqrt[n]{a^m} \).

So, in this case, \( (3xyz)^{\frac{3}{4}} \) converts back to:

\[
\sqrt[4]{(3xyz)^3}
\]

This matches the original form.

Would you like further clarification or details?

### Related Questions:
1. How do you simplify radical expressions involving variables?
2. What are the properties of exponents and radicals used in this transformation?
3. How do you handle fractional exponents in equations?
4. How do you multiply and divide expressions with fractional exponents?
5. How do you rationalize a denominator that contains radicals?

**Tip:** When converting between radical and exponential forms, remember that the index of the radical corresponds to the denominator of the exponent.

Convert the radical expression \(\sqrt[4]{(3xyz)^3}\) into exponential form and back into radical form.

Learn how to convert the radical expression \(\sqrt[4]{(3xyz)^3}\) into its exponential form and vice versa. This example demonstrates the core algebraic concepts involving radicals and exponents. Suitable for Grades 10-12 students.

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