Assume that all the variables represent positive numbers. Rewrite each expression so it does not contain any radicals,does not contain any negative exponents,does not contain any fractional exponents in the denominator and does not contain any fractional exponent greater than 1.

Root 5x^8y^4/z^3

To rewrite the expression \(\sqrt[5]{x^8 y^4 / z^3}\) without radicals, negative exponents, or fractional exponents in the denominator, follow these steps:

### Original expression:
\[
\sqrt[5]{\frac{x^8 y^4}{z^3}}
\]

1. **Convert the radical to fractional exponents**:  
   The fifth root of a term is the same as raising that term to the power of \( \frac{1}{5} \). So:
   \[
   \left( \frac{x^8 y^4}{z^3} \right)^{\frac{1}{5}} = \frac{x^{8 \cdot \frac{1}{5}} y^{4 \cdot \frac{1}{5}}}{z^{3 \cdot \frac{1}{5}}}
   \]

2. **Simplify the exponents**:
   \[
   \frac{x^{\frac{8}{5}} y^{\frac{4}{5}}}{z^{\frac{3}{5}}}
   \]

3. **Eliminate fractional exponents in the denominator**:  
   To do this, move the term with the fractional exponent in the denominator to the numerator with a negative exponent:
   \[
   x^{\frac{8}{5}} y^{\frac{4}{5}} z^{-\frac{3}{5}}
   \]

4. **Recombine terms into a single expression**:  
   You can write the expression as:
   \[
   x^{\frac{8}{5}} y^{\frac{4}{5}} z^{-\frac{3}{5}}
   \]

Thus, the expression without radicals, negative exponents, or fractional exponents in the denominator is:

\[
x^{\frac{8}{5}} y^{\frac{4}{5}} z^{-\frac{3}{5}}
\]

Do you have any questions or would you like further clarification?

Here are 5 related questions to expand on this topic:
1. How do you simplify an expression with fractional exponents?
2. What is the general rule for converting radicals to fractional exponents?
3. How can you simplify an expression involving negative exponents?
4. What is the meaning of raising a variable to a fractional power greater than 1?
5. How do you deal with higher roots, like cube roots or fifth roots, in algebraic expressions?

**Tip**: When simplifying expressions with radicals and exponents, always convert roots into fractional powers first—it makes further simplification much easier!

Rewrite the expression Root 5x^8y^4/z^3 so it does not contain any radicals, negative exponents, fractional exponents in the denominator, or fractional exponents greater than 1.

Learn how to simplify the expression √[5]{x^8 y^4 / z^3} by converting radicals to exponents, eliminating negative exponents, and avoiding fractional exponents in the denominator. This step-by-step guide is ideal for students in Grades 10-12, covering key algebraic techniques.

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